dent. library PRINCIPLES OF TRANSFORMER DESIGN * BY ALFRED STILL M.lNST.C.E., FEL.A,L!E., M.I.E.E. Professor of Electrical Design, Purdue University, Author of " Polyphase Currents," "Electric Power Transmission," etc. FIRST EDITION NEW YORK: JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED 1919 7 Engineering Library J COPYRIGHT, 1919, BY ALFRED STILL PHE88 OF BRAUNWORTH & CO. BOOK MANUFACTURERS BROOKLYN, N. V. PREFACE A BOOK which deals exclusively with the theory and design of alternating current transformers is not likely to meet the requirements of a College text to the same extent as if its scope were broadened to include other types of electrical machinery. On the other hand, the fact that there may be a limited demand for it by college students taking advanced courses in elec- trical engineering has led the writer to follow the method of presentation which he has found successful in teaching electrical design to senior students in the school of Electrical Engineering at Purdue University. Stress is laid on the fundamental principles of electrical engineering, and an attempt is made to explain the reasons underlying all statements and formulas, even when this involves the introduction of additional material which might be omitted if the needs of the practical designer were alone to be considered. A large portion of Chapter II has already appeared in the form of articles contributed by the writer to the Electrical World; but the greater part of the material in this book has not previously appeared in print. LAFAYETTE, IND. January, 1919 iii CONTENTS. PAGE PREFACE iii LIST OF SYMBOLS. . . ix CHAPTER I ELEMENTARY THEORY. TYPES. CONSTRUCTION ART. 1. Introductory i 2. Elementary Theory of Transformer 2 3. Effect of Closing the Secondary Circuit 6 4. Vector Diagrams of Loaded Transformer without Leakage. ... 10 5. Polyphase Transformers 12 6. Problems of Design 13 7. Classification of Alternating-current Transformers 14 8. Types of Transformers. Construction 17 9. Mechanical Stresses in Transformers 24 CHAPTER II INSULATION OF HIGH-PRESSURE TRANSFORMERS 10. The Dielectric Circuit 32 11. Capacity of Plate Condenser 40 12. Capacities in Series 42 13. Surface Leakage 46 14. Practical Rules Applicable to the Insulation of High-voltage Transformers 48 15. Winding Space Factor 51 16. Oil insulation 52 17. Terminals and Bushings , 54 18. Oil-filled Bushing 57 19. Condenser-type Bushing 62 v VI CONTENTS CHAPTER III EFFICIENCY AND HEATING OF TRANSFORMERS PAGE 20. Losses in Core and Windings 69 21. Efficiency % 73 22. Temperature of Transformer Windings 79 23. Heat Conductivity of Insulating Materials 80 24. Cooling Transformers by Air Blast 88 25. Oil-immersed Transformers, Self-cooling 91 26. Effect of Corrugations in Vertical Sides of Containing Tank ... 94 27. Effect of Overloads on Transformer Temperatures 98 28. Self-cooling Transformers for Large Outputs 103 29. Water-cooled Transformers 105 30. Transformers Cooled by Forced Oil Circulation 106 CHAPTER IV MAGNETIC LEAKAGE IN TRANSFORMERS. REACTANCE. REGULATION 31. Magnetic Leakage 107 32. Effect of Magnetic Leakage on Voltage Regulation 109 33. Experimental Determination of the Leakage Reactance of a Transformer 114 34. Calculation of Reactive Voltage Drop 117 35. Calculation of Exciting Current 1 25 36. Vector Diagram Showing Effect of Magnetic Leakage on Voltage Regulation of Transformers 132 CHAPTER V PROCEDURE IN TRANSFORMER DESIGN 37. The Output Equation 138 38. Specifications 140 39. Estimate of Number of Turns in Windings 141 40. Procedure to Determine Dimensions of a New Design 149 41. Space Factors : 151 42. Weight and Cost of Transformers 151 43. Numerical Example 154 CONTENTS Vli CHAPTER VI TRANSFORMERS FOR SPECIAL PURPOSES PAGE 44. General Remarks J 77 45. Transformers for Large Currents and Low Voltages i?7 46. Constant Current Transformers i?8 47. Current Transformers for use with Measuring Instruments 183 48. Auto-transformers JQ 1 49. Induction Regulators *97 LIST OF SYMBOLS ,4= area of equipotential surface perpendicular to lines of force (sq. cm.). A cross-section of iron in plane perpendicular to laminations (sq. in.). a = ampere- turns per inch length of magnetic path. a = total thickness of copper per inch of coil measured perpendicularly to layers. B = magnetic flux per sq. cm. (gauss). Bam is defined in Art. 9. b = total thickness of copper per inch of coil measured through insu- lation parallel with layers. C electrostatic capacity; or permittance, coulombs _ r ,r IN = =nux per unit e.m.f. (farad). volts Cmf = capacity in microfarads. c=a coefficient used in determining Vt. \fr D = flux density in electrostatic field =* -j = KkG (coulombs per sq. cm.). yl E= e.m.f. (volts), usually r.m.s. value, but sometimes used for max. value. 1= virtual value of induced volts in primary ( =E 2 Xjr] . E\ = component of impressed voltage to balance E\. EZ = secondary e.m.f. produced by flux <; induced secondary e.m.f. E e primary voltage equivalent to secondary terminal voltage (* IX X LIST OF SYMBOLS E p e.m.t. (volts) applied at primary terminals.. E s = secondary terminal voltage. -E z =irhpressed primary voltage when secondary is short-circuited. e=e.m.f. (volts). F= force (dynes). /= frequency (cycles per second). G=^ = potential gradient (volts per centimeter). g= distance between copper of adjacent primary and secondary coils, in centimeters (Fig. 42). H= magnetizing force, or m.m.f. per cm. h= length (cms.) defined in text (Fig. 42). 7 = r.m.s. value of current (amps.). /i = balancing component of primary current = I s ( ) . \Tp/ I c = current in the portion of an auto-transformer winding common to both primary and secondary circuits. Ie = total primary exciting current. /o = " wattless " component of I e (magnetizing component). Ip = total primary current. /o = total secondary current. /; = " energy " component of I e (" in-phase " component). J = 8.84Xio- 14 farads per cm. cube = the specific capacity of air. K 1 ' > definition follows formula (34) in Art. 27. -Ki>=kilovolts. k dielectric constant or relative specific capacity, or permittivity (k = i for air). & = heat conductivity (watts per inch cube per i c C.). k= coefficient used in calculating the effective cooling surface of corrugated tanks. k c = about i.SXio- 6 for copper. ki= (refer text (Art. 39) for definition). 1= length (cms.). /=mean length, in centimeters, of projecting end of transformer c,oil. 1 = length measured along line or tube of induction (cms.). ' LIST OF SYMBOLS xi ; c = mean length per turn of windings. /i = mean length of magnetic circuit measured along flux lines. M c = weight of copper in transformer coils (Ibs.) if o = weight of oil in transformer tank (Ibs.). W=27T/XlO~ 8 . n = - (in formula for calculating cooling surface of corrugated tanks). A = usually from 1.6 to 2 in B H (core loss formulas). P = weight of iron in transformer core (or portion of core), Ibs. p = thickness of half primary coil in centimeters (denned in text in connection with Fig. 42). R = resistance (ohms). Ri = resistance of primary winding (ohms). R 2 = resistance of secondary winding (ohms). Rh = " thermal ohms." IT P \ z R p equivalent primary resistance = Ri+R 2 [=- r = ratio ^ J-T T (auto-transformers). number of turns common to both circuits S = effective cooling surface of transformer tank (sq. in). 5 = thickness of half secondary coil (cms.) denned in text (Fig. 42). T = number of turns in coil of wire. Ti = number of turns in half primary group of coils adjacent to secondary coil. To = number of turns in half secondary group of coils adjacent to primary coil. Td = difference of temperature (degrees centigrade). To = initial oil temperature. T p = number of turns in primary winding. T s = number of turns in secondary winding. 7\ = oil temperature at end of time t m minutes. / = thickness (usually inches). t = interval of time (seconds). t m = interval of time (minutes). Vt = volts induced per turn of transformer winding. xii LIST OF SYMBOLS W = power (watts). W c = full-load copper loss (watts). Wt = core loss (watts). Wt = total transformer losses (watts). w = watts dissipated per sq. in. of (effective) tank surface. w = watts lost per Ib. of iron in (laminated) core. Xi = reactance (ohms) of one high-low section of winding. Xp = reactance (ohms) commonly referred to as equivalent primary reactance. ZD = impedance (ohms) on short circuit. = phase angle (cos 6 = power factor of external circuit). e = " electrical " angle (radians) = 2w ft. X = pitch of corrugations on tank surface. * = magnetic flux (Maxwells) in iron core. = phase angle (cos = power factor on primary side of transformer). SF = dielectric flux, or quantity of electricity, or electrostatic induc- tion = CE = A D coulombs. PRINCIPLES OF TRANSFORMER DESIGN CHAPTER I ELEMENTARY THEORY TYPES CONSTRUCTION 1. Introductory. The design of a small lighting transformer for use on circuits up to 2200 volts, or even 6600 volts, is a very simple matter. The items of importance to the designer are: (1) The iron and copper losses; efficiency, and tem- perature rise; (2) The voltage regulation, which depends mainly upon the magnetic leakage, and therefore upon the arrangement of the primary and secondary coils; (3) Economical considerations, including manufac- turing cost. With the higher voltages and larger units, not only does the question of adequate cooling become of greater importance; but other factors are introduced which call for considerable knowledge and skill on the part of the designer. The problems of insulation and pro- 2 PFIXCTPLES OF TRANSFORMER DESIGN tection against abnormal high-frequency surges in the external circuit are perhaps the most important; but with the increasing amount of power dealt with by some modern units, the mechanical forces exerted by the magnetic flux on short-circuits, or heavy over loads, may be enormous, requiring special means of clamping or bracing the coils, to prevent deformation and damage to insulation. Since we are concerned mainly with a study of the transformer from the view point of the designer, little will be said concerning the operation of transformers, or the advantages and disadvantages of the different methods of connecting the units on polyphase systems. It will, however, be necessary to discuss the theory underlying the action of all static transformers, and it is proposed to take up the various aspects of the subject in the following order : Elementary theory, omitting all considerations likely to obscure the fundamental principles; brief descrip- tion of leading types and methods of manufacture; problems connected with insulation; losses, heating, and efficiency; advanced theory, including study of magnetic leakage and voltage regulation; procedure in design; numerical examples of design; reference to special types of transformers. 2. Elementary Theory of Transformer. A single- phase alternating current transformer consists essen- tially of a core of laminated iron upon which are wound two distinct sets of coils, known as the primary and secondary windings, respectively, all as shown dia- grammatically in Fig. i. ELEMENTARY THEORY TYPES CONSTRUCTION 3 When an alternating e.m.f. of E v volts is applied to the terminals of the primary (P), this will set up a certain flux (<J>) of alternating magnetism in the iron core, and this flux will, in turn, induce a counter e.m.f. of self-induction in the primary winding; the action being similar to what occurs in any highly inductive coil or winding. Moreover, since the secondary coils ,1= 7olta Path of flux:J> maxwells linking with both windings. FIG. i. Essential Parts of Single-phase Transformer. although not in electrical connection with the pri- mary are wound on the same iron core, the variations of magnetic flux which induce the counter e.m.f. in the primary coils will, at the same time, generate an e.m.f. in the secondary winding. The path of the magnetic lines is usually through a closed iron circuit of low reluctance, in order that 4 PRINCIPLES OF TRANSFORMER DESIGN the exciting ampere- turns shall be small. There will always be some flux set up by the primary which does not link with the secondary, but the amount of this leakage flux is usually very small, and in any case it is proposed to ignore it entirely in this preliminary study. In this connection it may be pointed out that the design indicated in Fig. i, with a large space for leakage flux between the primary and secondary coils, would be unsatisfactory in practice; but the assump- tion will now be made that the whole of the flux ($ maxwells) which passes through the primary coils, links also with all the secondary coils. In other words, the e.m.f. induced in the winding per turn of wire will he the same in the secondary as in the primary coils. . Suppose, in the first place, that the two ends of the primary winding are connected to constant pres- sure mains, and that no current is taken from the secondary terminals. The total flux of <i> maxwells increases twice from zero to its maximum value, and decreases twice from its maximum to zero value, in the time of one complete period. The flux cut per second is therefore 4$/, and the average value of the induced e.m.f. in the primary is, _ -'-'average Tr >8 OilS, where T p stands for the number of turns in the primary wimding. If we assume the flux variations to be sinusoidal, the ELEMENTARY THEORY TYPES CONSTRUCTION form factor is i.n, and the virtual value of the induced primary volts will be, I0 8 The vector diagram corresponding to these condi- tions has been drawn in Fig. 2. Here OB represents the phase of the flux which is set up by the current OI e in \ FIG. 2. Vector Diagram o'f Unloaded Transformer. the primary. This total primary exciting current can be thought of as consisting of two components: the " wattless " component 01 'o which is the true magnetiz- ing current, in phase with the flux; and OI W (which owes its existence to hysteresis and eddy current losses) exactly 90 in advance of the flux. The volts induced in the primary are OE\ drawn 90 behind OB to repre- sent the lag of a quarter period. The voltage that must be impressed at the terminals of the primary is OE P made up of the component OE\ exactly equal but opposite to OEi, and E' \E V drawn parallel to 01 e and 6 PRINCIPLES OF TRANSFORMER DESIGN representing the IR drop in the primary circuit. The actual magnitude of this component would be I e Ri where R\ is the ohmic resistance of the primary; but in practice this ohmic drop is usually so small as to be negligible, and the impressed voltage E p is virtually the same as E'i, i.e., equal in amount, but opposite in phase to the induced voltage E\. For preliminary calculations it is, therefore, usually permissible to substitute the terminal voltage for the induced voltage, and write for formula (i) > (approximately). . . (10) Similarly, E.s= J ~~- - (approximately), . . (ib) where E s and T s stand respectively for the secondary terminal voltage and the number of turns in secondary. It follows that, 1? T (2) E 9 T p E s T s ' which is approximately true in all well-designed static transformers when no current, or only a very small current, is taken from the secondary. 3. Effect of Closing the Secondary Circuit. When considering the action of a transformer with loaded secondary, that is to say, with current taken from the secondary terminals, it is necessary to bear in mind that except for the small voltage drop due to ohmic resist- ance of the primary winding the counter e.m.f. induced ^LEMFNTARY THEORY TYPES CONSTRUCTION 7 by the alternating magnetic flux in the core must still be such as to balance the e.m.f. impressed at primary terminals. It follows that, with constant line voltage, the flux <f> has very -nearly the same value at full load as at no load. The m.m.f. due to the current in the sec- ondary windings would entirely alter the magnetization of the core if it were not immediately counteracted by a current component in the primary windings of exactly the same magnetizing effect, but tending at every instant to set up flux in the opposite .direction. Thus, in order to maintain the flux necessary to produce the required counter e.m.f. in the primary, any tendency on the part of the secondary current to alter this flux is met by a flow of current in the primary circuit; and since, in' well-designed transformers, the magnetizing current is always a small percentage of the full-load current, it follows that the relation I.T^LT,, (3) is approximately correct. Thus, !, is J- p where I p and I s stand respectively for the total primary and secondary current. The open-circuit conditions are represented in Fig. 3 where E v is the curve of primary impressed e.m.f. and L is the magnetizing current, distorted by the hysteresis of the iron core, as will be explained later. E s is the 8 PRINCIPLES OF TRANSFORMER DESIGN curve of secondary e.m.f. which coincides in phase with the primary induced e.m.f. and is therefore if we neglect the small voltage drop due to ohmic resistance of the primary exactly in opposition to the impressed e.m.f. The curve of magnetization (not shown) would FIG. 3. Voltage and Current Curves of Transformer with Open Second- ary Circuit. be exactly a quarter period in advance of the induced, or secondary, e.m.f. In Fig. 4, the secondary circuit is supposed to be closed on a non-inductive load, and the secondary current, I s will, therefore, be in phase with the secondary e.m.f. ELEMENTARY THEORY TYPES CONSTRUCTION 9 The tendency of the secondary current being to pro- duce a change in the magnetization of the core, the cur- rent in the primary will immediately adjust itself so as to maintain the same (or nearly the same) cycle of mag- netization as on open circuit; that is to say, the flux FIG. 4. Voltage and Current Curves of Transformer on Non-inductive Load. will continue to be such as will produce an e.m.f. in the primary windings equal, but opposite, to the primary impressed potential cjifference. The new curve of pri- mary current, I v (Fig. 4), is therefore obtained by adding the ordinates of the current curve of Fig. 3 to those of another curve exactly opposite in phase to the secondary 10 PRINCIPLES OF TRANSFORMER DESIGN current, and of such a value as to produce an equal mag- netizing effect. 4. Vector Diagrams of Loaded Transformer Without Leakage. The diagram of a transformer with secondary closed on a non-inductive load is shown in Fig. 5. In order to have a diagram of the simplest kind, not only the leakage flux, but also the resistance of the windings El I> , I u I. E 2 FIG. 5. Vector Diagram of Transformer on Non-inductive Load. will be considered negligible. The vectois then have the following meaning: QB = Phase of flux $ linked with both primary and secondary windings; I e = Exciting current necessary to produce flux 3>; Ez Secondary e.m.f. produced by alternations of the flux $; E'i= Primary e.m.f. equal, but opposite, to the e.m.f. produced by alternations of the flux $ (In this case it is equal to the applied e.m.f., since the IR drop is negligible) ; ELEMENTARY THEORY TYPES CONSTRUCTION 11 I s = Current drawn from secondary ; in phase with 2 ; /i = Balancing component of primary current, drawn T exactly opposite to I s and of value / s X^r; ? IP = Total primary current, obtained by combining /i with I e . In Fig. 6 the vectors have the same meaning as above, but the load is supposed to be partly inductive, which accounts for the lag of I s behind 2- FIG. 6. Vector Diagram of Transformer on Inductive Load. It is convenient in vector diagrams representing both primary and secondary quantities to assume a i : i ratio in order that balancing vectors may be drawn of equal length. The voltage vectors may, if preferred, be considered as wits per turn, while the secondary current vector can be expressed in terms of the pri- mary current by multiplying the quantity representing T the actual secondary current by the ratio ~. L v 12 PRINCIPLES OF TRANSFORMER DESIGN 5. Polyphase Transformers. Although we have con- sidered only the single-phase transformer, all that has been said applies also to the polyphase transformer because each limb can be considered separately and treated as if it were an independent single-phase trans- former. In practice it is not unusual to use single-phase trans- formers on polyphase systems, especially when the units are of very large size. Thus, in the case of a three-phase transmission, suppose it is desired to step up from 6600 volts to 100,000 volts, three separate single-phase trans- formers can be used, with windings grouped either Y or A, and the grouping on the secondary side need not necessarily be the same as on the primary side. A saving in weight and first cost may be effected by com- bining the magnetic circuits of the three transformers into one. There would then be three laminated cores each wound with primary and secondary coils and joined together magnetically by suitable laminated yokes ; but since each core can act as a return circuit for the flux in the other two cores, a saving in the total weight of iron can be effected. Except for the material in the yokes, this saving is similar to the saving of copper in a three-phase transmission line using three conductors only (as usual) instead of dx, as would be necessary if the three single-phase circuits were kept separate. In the case of a two-phase transformer, the windings would be on two limbs, and the common limb for the return flux need only be of sufficient section to carry \/2 times the flux in any one of the wound limbs. It is not always desirable to effect a saving in first ELEMENTARY THEORYTYPES CONSTRUCTION 13 cost by installing polyphase tiansformers in place of single-phase units, especially in the large sizes, because, apart from the increased weight and difficulty in hand- ling the polyphase transformer, the use .of single-phase, units sometimes leads to a saving in the cost of spares to be carried in connection with an important power devel- opment. It is unusual for all the circuits of a polyphase system to break down simultaneously, and one spare single-phase transformer might be sufficient to prevent a serious stoppage, while the repair of a large polyphase transformer is necessarily a big undertaking. 6. Problems in Design. The volt-ampere input of a single-phase transformer is E P I P , and if we substitute for E p the value given by formula (ia), we have A A.A.f Volt-amperes = -~ X $ X TJ P . Thus, for a given flux <, which will determine the cross- section of the iron core, there is a definite number of ampere turns which will determine the cross-section of the winding space. There is no limit to the number of designs which will satisfy the requirements apart from questions of heating and efficiency; but there is obvi- ously a relation between the weight of iron and weight of copper which wjll produce the most economical design, and this point will be taken up when discussing procedure in design. It will, however, be necessary to consider, in the first place, a few practical points in connection with the construction o;' transformers, and also the effect of insulation on the space available for the copper. The predetermination of the losses in both iron 14 PRINCIPLES OF TRANSFORMER DESIGN and copper must then be studied with a view to calcu- lating the temperature rise and efficiency. Finally, the flux leakage must be determined with a reasonable degree of accuracy because this, together with the ohmic resistance of the windings, will influence the voltage regulation, which must usually be kept within specified limits. 7. Classification of Alternating-current Transformers. Since we are mainly concerned with so-called constant- potential transformers as used on power and lighting circuits, we shall not at present consider constant-current transformers as used on series lighting systems and in connection with current-measuring instruments; neither shall we discuss in this place the various modifications of the normal type of transformer which render it avail- able for many special purposes. Transformers might be classified according to the method of cooling, or according to the voltage at the terminals, or, again, according to the number of phases of the system on which they will have to operate. Methods of cooling will be referred to again later when treating of losses and temperature rise; but, briefly stated, they include: (1) Natural cooling by air. (2) Self-cooling by oil; whereby, the natural circula- tion of the oil in which the transformer is immersed car- ries the heat to the sides of the containing tank. (3) Cooling by water circulation: a method generally similar to (2) except that coils of pipe carrying running water are placed near the top of the tank below the surface of the oil. ELEMENTARY THEORY TYPES CONSTRUCTION 15 (4) Cooling with forced circulation of oil: a method used sometimes when cooling water is not available. It permits of the oil being passed through external pipe coils having a considerable heat-radiating surface. (5) Cooling by air blast; whereby a continuous stream of cold air is passed over the heated surfaces, exactly as in the case of large turbo-generators. In regard to difference of voltage, this is mainly a matter of insulation, which will be taken up in Chap. II. The essential features of a potential transformer are the same whether the potential difference at ter- minals is large or small, but the high-pressure trans- former will necessarily occupy considerably more space than a low-pressure transformer of the same k.v.a. output. The difficulties of avoiding excessive flux leak- age and consequent bad voltage regulation are increased with the higher voltages. Low-voltage transformers are used for welding metals and for any purpose where very large currents are nec- essary, as for instance, in thawing out frozen water pipes, while transformers for the highest pressures are used for testing insulation. Testing-transformers to give up to 500,000 volts at secondary terminals are not uncommon, while one transformer (at the Panama- Pacific Exposition of 1915) was designed for an output of 1000 k.v.a. at 1,000,000 volts. This transformer weighed 32,000 lb., and 225 bbl. of oil were required to fill the tank in which it was immersed. A classification of transformers by the number of phases would practically resolve itself so far as present- day tendencies are concerned into a division between 16 PRINCIPLES OF TRANSFORMER DESIGN single-phase and three-phase transformers. From the point of view of the designer, it will be better to consider the use to which the transformer whether single-phase or polyphase will be put. This leads to the two classes: (1) Power transformers. (2) Distributing transformers. Power Transformers. This term is here used to include all transformers of large size as used in central generating stations and sub-stations for transforming the voltage at each end of a power transmission line. They may be designed for maximum efficiency at full load, because they are usually arranged in banks, and can be thrown in parallel with other units or discon- nected at will. Artificially cooled transformers of the air-blast type are easily built in single units for outputs of 3000 k.v.a. single-phase and 6000 k.v.a. three-phase; but the terminal pressure of these transformers rarely exceeds 33,000 volts. A three-phase unit of the air- blast type with 14,000 volts on the high-tension wind- ings has actually been built for an output of 20,000 k.v.a. For higher voltages the oil insulation is used, generally with water cooling-pipes. These transformers have been built three-phase up to 10,000 k.v.a. output from a single unit, for use on transmission systems up to 150,000 volts.* With the modern demand for larger *The 10,000 k.v.a. three-phase, 6600 to no,ooo-volt units in the power houses of the Tennessee Power Company on the Ocoee River weigh about 200,000 lb.; they are 19 ft. high, and occupy a floor space 20 ft. by 8 ft. Single-phase, oil-insulated, water-cooled transformers for a frequency of 60 cycles and a ratio of 13,200 to 150^000 volts have been built for an output of 14,000 k.v.a. from a single unit. ELEMENTARY THEORY TYPES CONSTRUCTION 17 transformers to operate out of doors, power transformers of the oil-immersed self-cooling type (without water coils) are now being constructed in increasing number. A self-cooling 25-cycle transformer for 8000 k.v.a. out- put has actually been built: a number of special tube- type radiators connected by pipes to the main oil tank are provided; the total cooling surface in contact with the air being about 7000 sq. ft. Distributing T"insformers. These are always of the self-cooling type, and almost invariably oil-immersed. They include the smaller sizes for outputs of i to 3 k.w. such as are commonly mounted on pole tops. These transformers are rarely wound for pressures exceeding 13,000 volts, the most common primary voltage being 2200. In the design of distributing transformers, it is neces- sary to bear in mind that since they are continuously on the circuit, the " all-day ' : losses which consist largely of hysteresis and eddy-current losses in the iron- must be kept as small as possible. In other words, it is not always desirable to have the highest efficiency at full load. 8. Types of Transformers. Construction. All trans- formers consist of a magnetic circuit of laminated iron with which the electric circuits (primary and secondary) are linked. A distinction is usually made between core- type and 'shell-type transformers. Single-phase trans- formers of the core- and shell-types are illustrated by Figs. 7 and 8, respectively. The former shows a closed laminated iron circuit two limbs of which carry the wind- ings. Each limb is wound with both primary and 18 PRINCIPLES OF TRANSFORMER DESIGN secondary circuits in order to reduce the magnetic leak- age which would otherwise be excessive. The coils may be cylindrical in form and placed one inside the other with the necessary insulation between them, or the wind- ings may be " sandwiched," in which case flat rect- angular or circular coils, alternately primary and sec- FIG. 7. Core-type Transformer. FIG. 8. Shell-type Transformer. ondary, are stacked one above the other with the requi- site insulation between. Fig. 8 shows a single set of windings on a central laminated core which divides after passing through the coils and forms what may be thought of as a shell of iron around the copper. The manner in which the core is usually built up in a large shell-type transformer is shown in Fig. 9. The thickness of the laminations ELEMENTARY THEORY TYPES CONSTRUCTION 19 varies between 0.012 and 0.018 in., the thicker plates being permissible when the frequency is low. A very usual thickness for transformers working on 25- and bo- cycle circuits is 0.014 in. The arrangement of the stampings is reversed in every layer in order to cover the joints and so reduce the magnetizing component of the primary current. A very thin coating of varnish FIG. 9. Method of Assembling Stampings in Shell-type Transformer. or paper is sufficient to afford adequate insulation be- tween stampings. Ordinary iron of good magnetic quality may be used for transformers on the lower fre- quencies, but it is customary to use special alloyed iron for 6o-cycle transformers. This material has a high electrical resistance and, therefore, a small eddy- current loss. The loss through hysteresis is also small, but the permeability of alloyed iron is lower than that of ordinary iron and this tends to increase the magnetiz- 20 PRINCIPLES OF TRANSFORMER DESIGN ing current. The cost of alloyed iron is appreciably higher than that of ordinary transformer iron. The choice of type whether " core " or " shell " will not greatly affect the efficiency or cost of the trans- former. As a general rule, the core type of construc- tion has advantages in the case of high-voltage trans- formers of small output, while the shell type is best 'adapted for low- voltage transformers of large output. Fig. 10 illustrates a good practical design of shell- type transformer in which a saving of material is effected by arranging the magnetic circuit to surround all four sides of a square coil. The dimensions of the iron cir- cuit, as indicated on the sketch, show a cross-section of the magnetic circuit outside the coils exactly double the cross-section inside the coils. This will be found to lead to slightly higher efficiency, for the same cost of material, than if the section were the same inside and outside the coil. It is generally advantageous to use higher flux densities in the iron upon which the coils are wound than in the remainder of the magnetic cir- cuit, because the increased iron loss is compensated for by the reduced copper loss due to the shorter average length per turn of the windings. Fig. ii illustrates a similar design of shell- type trans- former in which the magnetic circuit is still further divided, and the windings are in the form of cylindrical coils. The relative positions of primary and secondary coils need not be as shown in Figs. 10 and n, as they can be of the " pancake " shape of no great thickness, with primary and secondary coils alternating. A proper arrangement of the coils is a matter of great importance ELEMENTARY THEORY TYPES CONSTRUCTION 21 when it is desired to have as small a voltage drop as possible under load; but this point will be taken up FIG. 10. Shell-type Transformer with Distributed Magnetic Circuit. (Square core and coil.) again when dealing with magnetic leakage and regula- tion. Fig. 12 illustrates a common arrangement of the 22 PRINCIPLES OF TRANSFORMER DESIGN stampings and windings in a three-phase core-type transformer. Each of the three cores carries both pri- mary and secondary coils of one phase. The portions FIG. ii. Shell-type Transformer with Distributed Magnetic Circuit. (Berry transformer with circular coil.) of the magnetic circuit outside the coils must be of sufficient section to carry the same amount of flux as the wound cores. This will be understood if a vector diagram is drawn showing the flux relations in the ELEMENTARY THEORY TYPES CONSTRUCTION 23 various parts of the magnetic circuit. This use of cer- tain parts of the magnetic circuit to carry the flux com- mon to all the cores leads to a saving in material on what would be necessary for three single-phase trans- formers of the same total k.v.a. output; but, as men- FIG. 12. Three-phase Core-type Transformer. tioned in Article 5, it does not follow that a three-phase transformer is always to be preferred to three separate single-phase transformers. Figs. 13 and 14 show sections through three-phase transformers of the shell type. The former is the more 24 PRINCIPLES OF TRANSFORMER DESIGN common design, and it has the advantage that rect- angular shaped stampings can be used throughout. The vector diagram in Fig. 13 shows how the flux 3> e in the portion of the magnetic circuit between two sets of coils has just half the value of the flux $ in the cen- tral core. VECTOR DIAGRAM SHOWING THAT FIG. 13. Section through Three-phase Shell Transformer, phase consists of one H.T. and two L.T. coils.) (Each 9. Mechanical Stresses in Transformers. The mechanical features of transformer design are not of sufficient importance to warrant more than a brief discussion. In the smaller transformers it is merely necessary to see that the clamps or frames securing the stampings and coils in position are sufficiently sep- arated from the H.T. windings, and that 'bolts in which ELEMENTARY THEORY TYPES CONSTRUCTION 25 e.m.f.'s are likely to be generated by the main or stray magnetic fluxes are suitably insulated to prevent the establishment of electric currents with consequent PR losses. The tendency in all modern designs is to avoid cast iron, and use standard sections of structural steel in the assembly of the complete transformer. In this FIG. 1 4. Special Design of Three-phase Shell- type Transformer. manner the cost of special patterns is avoided and a saving in weight is usually effected. The use of stand- ard steel sections also gives more flexibility in design, as slight modifications can be made in dimensions with very little extra cost. In large transformers, the magnetic forces exerted under conditions of heavy overloads or short-circuits 26 PRINCIPLES OF TRANSFORMER DESIGN may be sufficient to displace or bend the coils unless these are suitably braced and secured in position; and since the calculation of the stresses that have to be resisted belong properly to the subject of electrical design, it will be necessary to determine how these stresses can be approximately predetermined. The absolute unit of current may be defined as the current in a wire which causes one centimeter length of the wire, placed at right angles to a magnetic field, to be pushed sidewise with a force of one dyne when the density of the magnetic field is one gauss. Since the ampere is one- tenth of the absolute unit of current, we may write, ' BIl where F = Force in dynes; B = Density of the magnetic field in gausses; / = Current in the wire (amperes) ; / = Length of the wire (centimeters) in a direction perpendicular to the magnetic field. It follows that the force tending to push a coil of wire of T turns bodily in a direction at right angles to a uniform magnetic field of B gausses (see Fig. 15) is 17 BITl A p = _ dynes. 10 If both current and magnetic field are assumed to vary periodically according to the sine law, passing through corresponding stages of their cycles at the ELEMENTARY THEORY TYPES-CONSTRUCTION 27 same instant of time, we have the condition which is approximately reproduced in the practical transformer where the leakage flux passing through the windings is due to the currents in these windings. Coil of T wires, each -carrying I amperes FIG. 15. Force Acting on Coil-side in Uniform Magnetic Field. Since the instantaneous values of the current and flux density will be 7 max sin 0, and B max sin 8, respectively, the average mechanical force acting upon the coil may be written, Tl. i C* . TIL average = /max^max I sill 10 if*. Sll KjQ .2 fiflfi 10X2 dynes. 28 PRINCIPLES OF TRANSFORMER DESIGN If the flux density is not uniform throughout the sec- tion of coil considered, the average value of 5 max should be taken. Let this average value of the maximum den- sity be denoted by the symbol B am . Then, since i Ib. = 444,800 dynes, the final expression for the average force tending to displace the coil is, 1 II m ax -Dam .. Force= w;~ lb In large transformers the amount of leakage flux passing through the coils may be considerable. It will be very nearly directly proportional to / max , and the mechanical forces on transformer coils are therefore approximately proportional to the square of the current. As the short-circuit current in a transformer which is not specially designed with high reactance might be thirty times the normal full-load current, the mechan- ical forces due to a short-circuit may be about 1000 times as great as the forces existing under normal work- ing conditions. Except in a few special cases, the calculation of the leakage flux is not an easy matter, and the value of B &m in Eq. (4) cannot usually be predetermined exactly; but it can be estimated with sufficient accuracy for the purpose of the designer, who requires merely to know approximately the magnitude of the mechanical forces which have to be resisted by proper bracing of the coils. The calculation of leakage flux will be considered when discussing voltage regulation; but in the case of "sandwiched " coils as, for instance, in the shell type of ELEMENTARY THEORY TYPES CONSTRUCTION 29 :ransformer shown in Fig. 16, the distribution of the leakage flux will be generally as indicated by the dia- gram plotted over the coils at the bottom of the sketch. -m Max. T7 Bam FIG. 1 6. Forces in Transformer Coils Due to Leakage Flux. When the relative directions of the currents in the primary and secondary coils are taken into account, it 30 PRINCIPLES OF TRANSFORMER DESIGN will be seen that all the forces tending to push the coilr sidewise are balanced, except in the case of the two outside coils. In each individual coil the effect of the leakage flux is to crush the wires together; but the end FIG. 17. Core-type Transformer with " Sandwiched " Coils. coils will be pushed outward unless properly secured in position. Since there is no resultant force tending to move the windings bodily relatively to the iron stampings, a simple form of bracing consisting of insulated bars and ELEMENTARY THEORY TYPES CONSTRUCTION 31 tie rods, as shown in Fig. 16 will satisfy all requirements, and this bracing can be quite independent of the frame- work or clamps supporting the transformer as a whole. In the case of core-type transformers, with rect- angular coils arranged axially one within the other, the mechanical forces will tend to force the coils into a cir- cular shape. With cylindrical concentric coils, no spe- cial bracing is necessary provided the coils are symmet- rically placed axially; but if the projection of one coil beyond the other is not the same at both ends, there will be an unbalanced force tending to move one coil axially relatively to the other. If the core type of transformer is built up with flat strip " sandwiched " coils, the problem is generally similar to that of the shell type of construction. A method of securing the end coils in position with this arrangement of windings is illus- trated by Fig, 17, CHAPTER II INSULATION OF HIGH-PRESSURE TRANSFORMERS 10. The Dielectric Circuit. Serious difficulties are not encountered in insulating machinery and apparatus for working pressures up to 10,000 or 12,000 volts, but for higher pressures (as in 150,000- volt transformers) designers must have a thorough understanding of the dielectric circuit,* if the insulation is to be correctly and economically proportioned. The information here assembled should make the fundamental principles of insulation readily understood and should enable an engineer to determine in any specific design of trans- former the thicknesses of insulation required in any particular position, as between layers of windings, between high-tension and low- tension coils, and be- tween high-tension coils and grounded metal. The data and principles outlined should also facilitate the deter- mination of dimensions and spacings of high-tension terminals and bushings of which the detailed design is usually left to specialists in the manufacture of high- tension insulators. In presenting this information two questions are considered: (i) What is the dielectric * " Insulation and Design of Electrical Windings," by A. P. M Fleming and R. Johnson Longmans, Green & Co. " Dielectric Phenomena in High- voltage Engineering," by F. W. Peek, Jr. McGraw-Hill Book Company, Inc. 32 INSULATION OF HIGH-PRESSURE TRANSFORMERS 33 strength of the insulating materials used in transformer design? and (2) how can the electric stress or voltage gradient be predetermined at all points where it is liable to be excessive? Apart from a few simple problems of insulation capable of a mathematical solution, the chief difficulty encountered in practice usually lies in determining the distribution of the dielectric flux, the concentration of which at any particular point may so increase the fkix density and the corresponding electric stress that dis- ruption of the dielectric may occur. The conception of lines of dielectric flux, and the treatment of the dielec- tric circuit in the manner now familiar to all engineers in connection with the magnetic circuit has made it pos- sible to treat insulation problems * in a way that is equally simple and logical. The analogy between the dielectric and magnetic circuits may be illustrated by Fig. 18, where a metal sphere is supposed to be placed some distance away from a flat metal plate, the intervening space being occupied by air, oil, or any insulating substance of constant specific capacity. This arrangement constitutes a con- denser of which the capacity is (say) C farads. If a difference of potential of E volts is established between * The dielectric circuit is well treated from this point of view in the following (among other) books: "The Electric Circuit," by V. Karapetoff McGraw-Hill Book Company, Inc. " Electrical Engineering," by C. V. Christie McGraw-Hill Book Company, Inc. " Advanced Electricity and Magnetism," by W. S. Franklin and B. MacNutt Macmillan Company. 34 PRINCIPLES OF TRANSFORMER DESIGN the sphere and the plate, the total dielectric flux, ^ will have to satisfy the equation * = EC, (5) where ^ is expressed in coulombs, E in volts, and C in farads. The quantity M> coulombs of electricity should not be considered as a charge which has been carried from the sphere to the plate on the surface of which it remains, because the whole of the space occupied by the dielectric is actually in a state of strain, like a deflected spring, ready to give back the energy stored in it when the potential difference causing the deflection or displace- ment is removed. Instead, the dielectric should be considered as an electrically elastic material which will not break down or be ruptured until the " elastic limit '' has been reached. The quantity SF, which is called the dielectric flux, may be thought of as being made up of a definite number of unit tubes of induction, the direc- tion of which in the various portions of the dielectric field is represented by the full lines in Fig. 18. The name of the unit tube of dielectric flux is the coulomb. If the sphere were the north pole and the plate the south pole of a magnetic circuit, the distribution of flux lines would be similar. The total flux would then be denoted by the symbol $, and the unit tube of induc- tion would be called the maxwell. In place of formula (5) the following well-known equation could then be written : < = Mmf X permeance (6) INSULATION OF HIGH-PRESSURE TRANSFORMERS 35 This expression is analogous, to the fundamental equation for a dielectric circuit, the electrostatic capacity C being, in fact, a measure of the permeance of the di- electric circuit, while , sometimes called the elastance, C- may be compared with reluctance in the magnetic circuit. The dotted lines in Fig. 18 are sections through equi- potential surfaces. The potential difference between \ FIG. 1 8. Distribution of Dielectric Flux between Sphere and Flat Plate. any two neighboring surfaces, as drawn, is one-quarter of the total. At ,all points the lines of force, or unit tubes of induction, are perpendicular to the equipoten- tial surfaces. Furthermore, the flux density, or cou- lombs per square centimeter, through any small portion A of an equipotential surface over which the distribu- tion may be considered practically uniform is ''A' (7) 36 PRINCIPLES OF TRANSFORMER DESIGN The capacity, or permittance* of a small element of the dielectric circuit of length / and cross-section A A is proportional to --, or with the proper constants inserted, Electrostatic capacity = C = ( , lo9 -W farads (8) \47r(3Xio 10 ) 2 / / wherein the numerical multiplier results from the choice of units. The factor k is the specific inductive capacity, or dielectric constant, of the material (k = i in air), while the unit for / and A is the centimeter. This ex- pression for capacity may conveniently be rewritten as ' 8.84 kA . t mf = ^ microfarads. ... (9) Values of k are given in the accompanying table to- gether with the dielectric strengths of the materials. These figures are only approximate, those referring to dielectric strength merely serving as a rough indication of what the material of aveiage quality may be expected to withstand. The figures indicate the approximate virtual or r.m.s. value of the sinusoidal alternating voltage which, if applied between two large flat elec- trodes, would lead to the breakdown of a i-cm. slab of insulating material placed between the electrodes. What is generally understood by the disruptive gra- dient, or stress in kilovolts per centimeter, would be * The reciprocal of elastance. INSULATION OF HIGH-PRESSURE TRANSFORMERS 37 about \/~2 times the value given in the last column of the table. Thus, if a battery or continuous-current generator were used in the test, the pressure necessary to break down a 0.75-011. film of air between two large flat parallel plates would be loooXV^X 22X0. 75 = 23,400 volts. DIELECTRIC CONSTANT AND DIELECTRIC STRENGTH OF INSULATORS Material. Dielectric Constant, k. Dielectric Strength Kv. per Cm. Air j 22 Transformer oil 2 3 Paper (dry) 2 80 Paper (oil impregnated) IOO Pressboard (dry or varnished). . . 2 I2C Pressboard (oil impregnated) r I ?O Treated wood (across grain) 3 to 6 7O Porcelain Varnished cambric 4-5 r 110 2OO Mica. 6 to 8 6OO Micanite ^ to 7 3OO Glass 4 to 10 oo Conductors Infinity Returning to Formula (7), let electric flux or quantity of electricity, ^, be expressed in terms of capacity and e.m.f., with a view to determining the relation between flux density and electric stress. The Formula (8) may be written C=Kkj farads, 38 PRINCIPLES OF TRANSFORMER DESIGN where K stands for the numerical constant. Sub- stituting in Formula (5), whence Since is the potential gradient, or voltage drop / per centimeter, which is sometimes referred to as the electrostatic force or electrifying force, and denoted by the symbol G, we may write, D = KkxG. . . , . . (10) The analogous expression for the magnetic circuit is, In the case of a dielectric circuit, electric flux density = e.m.f. per centimeter X " conductivity " of the material to dielectric flux, while in the magnetic circuit, magnetic flux density = m.m.f. per centimeter X " conductivity " of the material to magnetic flux. Since the electric stress or voltage gradient G is directly proportional (in a given material) to the flux density D, it follows that when the concentration of the flux tubes is such as to produce a certain maximum density at any point, breakdown of the insulation will occur at this point. Whether or not the rupture will extend entirely through the insulation will depend upon INSULATION OF HIGH-PRESSURE TRANSFORMERS 39 the value of the flux density (consequently the potential gradient) immediately beyond the limits of the local breakdown. Given two electrical conductors of irregular shape, separated by insulating materials, the problem of cal- culating the capacity of the condenser so formed is very similar to that of calculating the permeance of the magnetic paths between two pieces of iron of very high permeability separated by materials of low per- meability. There is no simple mathematical solution to such a problem, and the best that can be done is to fall back on the well-established law of maximum per- meance, or " least resistance." According to this law the lines of force and equipotential surfaces will be so shaped and distributed that the permittance, or capacity, of the flux paths will be a maximum. With a little experience, ample time, and a great deal of patience, the probable field distribution can generally be mapped out, even in the case of irregularly shaped surfaces, with sufficient accuracy to emphasize the weak points of the design and to permit of the maximum voltage gradient being approximately determined.* Before illustrating the application of the above prin- ciples in the design of transformer insulation, it will be advisable to assemble and define the quantities which are of interest to the engineer in making practical calculations. * This method of plotting flux lines is explained, in connection with the magnetic field, at some length in the writer's book " Principles of Electrical Design." McGraw-Hill Book Co., Inc. 40 PRINCIPLES OF TRANSFORMER DESIGN Symbol: E, e = e.m.f . or potential difference (volts) ; / = length, measured along line of force (centimeters) ; yl=Area of equipotential surface perpendicular to lines of force (square centimeters) ; de G = -r, = potential gradient (volts per centimeter); C = Capacity or permittance (farads) ; coulombs (farads = ,- = nux per unit e.m.f.); K constant = 8.84 X io~ 14 (farads per centimeter cube, being the specific capacity of air) ; k = dielectric constant, or relative specific capacity, or permittivity (k i for air) ; ^ = dielectric flux, or electrostatic induction (^ = CE AD coulombs) ; ^ Z) = flux density = -r = KkG (coulombs per square centi- meter). 11. Capacity of Plate Condenser. Imagine two par- allel metal plates, as in Fig. 19, connected to the oppo- site terminals of a direct-current generator or battery. The area of each plate is A square centimeters and the separation between plates is / centimeters, the dielectric or material between the two surfaces being air. The edges of the plates should be rounded off to avoid con- centration of flux lines. If the area A is large in com- parison with the distance /, a uniform distribution of the flux ^ may be assumed in the air gap, the density being INSULATION OF HIGH-PRESSURE TRANSFORMERS 41 8.1 By Formula (9) the capacity is C m / = 10 mcro- farads, since the specific capacity of air (k) is i. As- suming numerical values, let A = 1000 sq. cm., and = o.5 cm. Then, C = - ~ = i.yyXio- 10 farads. 10 If - = 10,000 volts, the potential gradient will be G = = 20,000 volts per centimeter. There will be [Area lA^lOOOsq.cm. Distanced = 0.5 cm, FIG. 19. Flat Electrodes Separated by Air. no disruptive discharge, however, because a gradient of 31,000 volts per centimeter is necessary to cause break-down in air. By Formula (5) the total dielectric flux is ^ = 10,000 Xi.77Xio- 10 = i,77Xio- 6 coulombs. Charging Current with Alternating Voltage. The effect of an alternating e.m.f., the crest value of which is 10,000 volts, would be to displace the above quantity of elec- tricity 4/ times per second, / being the frequency. 42 PRINCIPLES OF TRANSFORMER DESIGN The quantity of electricity can be expressed in terms of current and time, thus, quantity = current X time, or coulombs = average value of current (in amperes) during quarter peri o d X time (in seconds) of one quarter period. / 2 V~2\ I Therefore, ^ = /X|- -) :., where / stands for the \ TT 747 virtual or r.m.s. value of the charging current on the sine wave assumption. Transposing terms, I = ~. 2V 2 If E is now understood to stand for the virtual value of the alternating potential difference, ty = CExv / 2, whence I = 2irfCE, which is the well-known formula for calculating capacity current on the assumption of sinusoidal wave shapes. 12. Capacities in Series. When condensers are con- nected in parallel on the same source of voltage, the total dielectric flux is evidently determined by summing up the fluxes as calculated or measured for the individual condensers. In other words, the total capacity is the sum of the individual capacities. With condensers in series, however, the total flux, or displacement, will be the same for all the capacities in series, therefore, the calculations may be simplified just as for electric or magnetic circuits by adding the reciprocals of the con- ductance or permeance. The conception of elastance, corresponding to resistance in the electric circuit and reluctance in the magnetic circuit, is thus seen to have certain advantages. In the dielectric circuit Elastance = permittance (or capacity) C' INSULATION OF HIGH-PRESSURE TRANSFORMERS 43 For a concrete example, assume that a 0.3-011. plate of glass is inserted between the electrodes of the con- denser shown in Fig. 19. The modified arrangement is illustrated by Fig. 20. On first thought it might appear that this arrangement would improve the insulation, but care must always be taken when putting layers of insulating materials of different specific inductive capac- ity in series, as this example will illustrate. In addition to the elastance of a 0.3 -cm. layer of glass there is the 0.3 cm. thick. FIG. 20. Electrodes Separated by Air and Glass. elastance of two layers of air of which the total thickness is 0.2 cm. Assuming that the value of the dielectric constant k for the particular quality of glass used is 7 and that G g and G a are the potential gradients in the glass and air respectively, then, by formula (10) KG a yKGo, whence G a jGg. Taking the total potential difference between elec- trodes as 10,000 volts, the same as used in considering Fig. 19, E = 10,000 = o.2G a +0.360, whence G g = 5880 volts 44 PRINCIPLES OF TRANSFORMER DESIGN per centimeter, and G a = 41,100 volts per centimeter. Such a high gradient as 41,100 would break down the layers of air and would manifest itself by a bluish elec- trical discharge between the metal plates and the glass. On the other hand, the gradient of 5880 volts per cen- timeter would be far below the stress necessary to rupture the glass. Nevertheless a discharge across air spaces should always be avoided in practical designs because of its injurious effect on the metal surfaces and also on certain types of insulating material. It should be observed that the introduction of the glass plate has appreciably increased the capacity of the con- denser. For example, with the same voltage (E = 10,000) as before, the total flux is now^ = A D = 1000 (8.84X io~ 14 X4i,ioo) =3.63Xio~ 6 coulombs. This value is about double the value calculated with only air between the condenser plates. As a practical application of the principles governing the behavior of condensers in series, consider the insu- lation between the coils and core of an air-cooled trans- former, i.e., of which the coils are not immersed in oil. In addition assume the insulation to consist of layers of different materials made up as follows: TOTAL 1 SICKNESS. Material. Mils.. Centi- meters. Dielectric Constant, k. Cotton braiding and varnished cambric Micanite . . 70 12=? 0.178 O 3I7 5 6 Pressboard 62 o 158 Air spaces (estimated) 24 o 061 i INSULATION OF HIGH-PRESSURE TRANSFORMERS 45 Then, suppose it is desired to determine how high an alternating voltage can be applied between the coils and the core before the maximum stress in the air spaces exceeds 31,000 volts per centimeter, the gradient which will cause disruption and static discharge, with the consequent danger to the insulation due to local heating and chemical action. Assuming the coil to constitute one flat plate of a condenser of which the other plate is the iron frame or core, the effect is that of a number of plate condensers in series the total elas- tance being - = + ++--. C C\ 2, Ca C By Formula (8), the individual capacities for the k same surface area are proportional to j, and = C ki J?2 k$ k Since KA KAE KE KE E C '' * = ~D~~KG ai r~Gair J the permissible maximum value of E is = 6260 volts (maximum). The r.m.s. value of the corresponding sinusoidal alter- 6260 natmg voltage is =-=4430, which is the limiting V2 46 PRINCIPLES OF TRANSFORMER DESIGN potential difference between windings and grounded metal work if the formation of corona is to be avoided. A transformer having insulation made up as previously described would be suitable for a 66oo-volt three-phase circuit with grounded neutral; but for higher voltages the insulation should be modified, or oil immersion should be employed to fill all air spaces. If the oil- cooled construction is employed, the previously con- sidered insulations (slightly modified in view of pos- sible action of the oil upon the varnish) would probably be suitable for working voltages up to 15,000. 13. Surface Leakage. A large factor of safety must be allowed when determining the distance between electrodes measured over the surface of an insulator. Whether or not spark-over will occur depends not only upon the condition of the surface (clean or dirty, dry or damp), but also upon the shape and position of the terminals or conductors. It is therefore almost impos- sible to determine, other than by actual test, what will happen in the case of any departure from standard practice. Surface leakage occurs under oil as well as in air, but generally speaking, the creepage distance under oil need be only about one-quarter of what is necessary in air. An important point to consider in connection with surface leakage is illustrated by Figs. 21 and 22. In Fig. 21, a thin disk of porcelain (or other solid insulator) separates the two electrodes, while in Fig. 22, the same material is in the form of a thick block providing a leakage path (/) of exactly the same length as in Fig. 21. The voltage required to cause spark-over will be con- INSULATION OF HIGH-PRESSURE TRANSFORMERS 47 siderably greater for the block of Fig. 22 than for the disk of Fig. 21. This condition exists because the flux concentration due to the nearness of the terminals in Fig. 21 begins breaking down the layers of air around the edges of the electrodes at a much lower total poten- tial difference than will be necessary in the case of the thicker block of Fig. 22. The effect of the incipient breakdown is, virtually, to make a conductor of the air FIG. 21. FIG. 22. FIG. 21. Surface Leakage over Thin Plate. FIG. 22. Surface Leakage Over Thick Insulating Block. around the edges of the metal electrodes, and a very slight increase in the pressure will often suffice to break down further layers of air and so result in a discharge over the edges of the insulating disk. The phenomenon- of so-called surface leakage may thus be considered as largely one of flux concentration or potential gra- dient. Sometimes it will be easier to eliminate trouble due to surface leakage by altering the design of ter- 48 PRINCIPLES OF TRANSFORMER DESIGN minals and increasing the thickness of the insulation than by adding to the length of the creepage paths. 14. Practical Rules Applicable to the Insulation of High-voltage Windings. For working pressures up to 16,000 volts, solid insulation, including cotton tape, micanite, pressboard, horn paper, or any insulating material of good quality used to separate the windings from the core or framework, should have a total thick- ness of approximately the following values: Voltage. Thickness of Insulation (Mils) IIO 4 400 45 1,000 65 2,200 90 6,600 180 12,000 270 16,000 350 In large high- volt age power transformers, cooled by air blast, the air spaces are relied upon for insulation. The clearances between coils and core or case are neces- sarily much larger than in oil-cooled transformers, and calculations similar to the example previously worked out should be made to determine whether or not the insulation is sufficient and suitably proportioned to prevent brush discharge. The calculations are made on the basis of several plate condensers in series; thus the flux density and dielectric stress in the various layers of insulation can be approximately predetermined. The difficulty of avoiding static discharges will generally INSULATION OF HIGH-PRESSURE TRANSFORMERS 49 stand in the way of designing economical air-cooled transformers for pressures much in excess of 30,000 volts. A rough rule ior air clearance is to allow a kv-j-i distance equal to inches, where kv stands for 4 the virtual value of the alternating potential differ- ence in kilovolts between the two surfaces considered. With oil-immersed transformers, the oil channels should be at least 0.25 in. wide in order that there may be free circulation of the oil. In high-voltage trans- formers having a considerable thickness of insulation between coils and core, it is advantageous to divide the oil spaces by partitions of pressboard or similar mate- rial. Assuming the total thickness of oil to be - no greater than that of the solid insulation, a safe rule is to allow i mil for every 25 volts. For instance, a total thickness of insulation of i in. made up of 0.5 in. of solid insulation and two 0.25 in. oil ducts would be suit- able for a working pressure not exceeding 25X1000 = 25,000 volts. Further particulars relating to oil insula- tion will be given later. It is customary to limit the volts per coil to 5000, and the volts between layers of winding to 400. Special attention must be paid to the insulation under the finishing ends of the layers by providing extra insula- tion ranging from thin paper to Empire cloth or even thin fullerboard, the material depending upon the voltage and also upon the amount of mechanical protection required to prevent cutting through the insulation where the wirer cross. Sometimes the insulation is bent around the end wires of a layer to prevent breakdown 50 PRINCIPLES OF TRANSFORMER DESIGN over the ends of the coil. Where space permits, however, the layers of insulation may be carried beyond the ends of the winding so as to avoid surface leakage. This arrange- ment is more easily carried out in core-type transformers than in shell-type units. A practical rule for deter- mining the surface distance (in inches) required to pre- vent leakage (given by Messrs. Fleming and Johnson in the book previously referred to) is "to allow 0.5 in. +o.5X kilovolts, when the surfaces are in air. For sur- faces under oil, the allowance may be 0.5+0.1 Xkilovolts. In any case it is important to see that the creepage sur- faces are protected as far as possible from deposits of dirt. When the coils of shell-type transformer are "sandwiched," it is customary to use half the normal number of turns in the low-tension coils at each end of the stack. This has the advantage of keeping the high- tension coils well away from the iron stampings and clamping plates or frame. Extra Insulation on End Turns. Concentration of potential between turns at the ends of the high-tension winding is liable to occur with any sudden change of voltage across the -transformer terminals, such as when the supply is switched on, or when lightning causes potential disturbances on the transmission lines. It is, therefore, customary to pay special attention to the insu- lation of the end turns of the high-tension winding. Transformers for use on high-voltage circuits usually have about 75 ft. at each end of the high-tension winding insulated to withstand three to four times the voltage between turns that would puncture the insulation in the body of the winding. INSULATION OF HIGH-PRESSURE TRANSFORMERS 51 It is very difficult to predetermine the extra pressure to which the end turns of a power transformer con- nected to an overhead transmission line may at times be subjected, but it is safe to say that the instantaneous potential difference between turns may occasionally be of the order of forty to fifty times the normal working pressure. In such cases the usual strengthening of the insulation on the end turns would not afford adequate protection, and for this reason a separate specially designed reactance coil connected to each end of the high- tension winding would seem to be the best means of guarding against the effects of surges or sudden changes of pressure occurring in the electric circuit outside the transformer. The theory of abnormal pressure rises in the end sections of transformer windings will not be discussed here. 15. Winding Space Factor. Knowing the thickness of the cotton covering on the wires, the insulation between layers of winding, between coil and coil and between coil and iron stampings, it becomes an easy matter to determine approximately the total cross-section of the winding-space to accom- modate a given cross-section of copper. The ratio cross-section of copper , which is known as the cross-section of winding space space factor, will naturally decrease with the higher voltages and smaller sizes of wire. This factor may be as high as 0.46 in large transformers for pressures not exceeding 2200 volts; in 33,000- volt transformers for outputs of 200 k.v.a. and upward it will have a value ranging between 0.35 and 0.2, while in oil-immersed 52 PRINCIPLES OF TRANSFORMER DESIGN power transformers for use on ioo,ooo-volt circuits the factor may be as low as 0.06. 16. Oil Insulation. There is a considerable amount of published matter relating to the properties of insulating oils, and also to the various methods of testing, puri- fying, and drying oils for use in transformers. A con- cise statement of the points interesting to those installing or having charge of transformers will be found in W. T. Taylor's book on transformers.* What follows here is intended merely as a guide to the designer in providing the necessary clearances to avoid spark-over, including a reasonable factor of safety. Mineral oil is generally employed for insulating pur- poses, its main function in transformers being to trans- fer the heat by convection from the hot surfaces to the outside walls of the containing case, or to the cooling coils when these are provided. The presence of an extremely small percentage of water reduces the insu- lating properties of oil considerably. It is therefore important to test transformer oil before using it, and if necessary extract the moisture by filtering through dry blotting paper, or by any other approved method. Dry oil will withstand pressures up to 50,000 volts (alter- nating) between brass disks 0.5 in. in diameter with a separation of 0.2 in. For use in high- voltage trans- formers, the oil should be required to withstand a test * " Transformer Practice," by W. T. Taylor McGraw-Hill Book Company, Inc. For further information refer H. W. Tobey on the "Dielectric Strength of Oil " Trans. A.I.E.E.; Vol. XXIX, page 1189 (1910). Also " Insulating Oils," Journ. Inst. E.E.. Vol. 54, page 497 (1916). INSULATION OF HIGH-PRESSURE TRANSFORMERS 53 of 45,000 volts under the above conditions. The good insulating qualities of oil suggest that only small clear- ances would be required in transformers, even for high voltages; but the form of the surfaces separated by the layer of oil will have a considerable effect upon the con- centration of flux density, and therefore upon the volt- age gradient. As an example, if 100,000 volts breaks down a i -in. layer of a certain oil between two parallel disks 4 in. in diameter, the same pressure will spark across a distance of about 3.5 in. between a disk and a needle point. Partitions of solid insulation such as pressboard or fullerboard are always advisable in the spaces occu- pied by the oil, since they will prevent the lining up of partly conducting impurities along the lines of force and reduce the total clearance which would otherwise be necessary. In a transformer oil of average quality, the sparking distance between a needle point and a flat plate is approx- imately (o.25+o.o4Xkv.) inches. Since there may be sharp corners or irregularities corresponding to a needle point, which will produce concentration of dielectric flux, it therefore seems advisable to introduce a factor of safety for oil spaces between high tension and grounded metal fgr instance, between the ends of high-tension coils and the containing case by basing the oil space dimension on the formula, Thickness of oil (inches) =0.25+0.08 Xkv., . (n) where kv. stands for the working pressure in kilovolts. 54 PRINCIPLES OF TRANSFORMER DESIGN With two or three partitions of solid insulating mate- rial dividing the oil space into sections, the total ness need not exceed 0.25+0.05 Xkv (12) If the total thickness of solid insulation is about equal to that of the oil ducts (not an unusual arrangement between coils and core), the rule previously given for solid insulation may be slightly modified to include a minimum thickness of 0.25 in., and put in the form, Total thickness of oil ducts plus ) solid insulation of app'roxi- I =0.2 5 +0.03 Xkv. (13) mately equal thickness (inches) J A suitable allowance for surface leakage under oil, in inches, as already given, is 0.5 +0.1 Xkv. . o, . . , (14) 17. Terminals and Bushings. The exact pressure which will cause the breakdown of a transformer ter- minal bushing generally has to be determined by test, because the shape and proportions of the metal parts are rarely such that the concentration of flux density at corners or edges can be accurately predetermined.* * The reader who desires to go deeply into the study of high-pressure terminal design should refer to the paper by Mr. Chester W. Price entitled " An Experimental Method of Obtaining the Solution of Elec- trostatic Problems, with Notes on High-voltage Bushing Design." Trans. A.I.E.E., Vol. 36, page 905 (Nov., 1917). INSULATION OF HIGH-PRESSURE TRANSFORMERS 55 However, there are certain important points to bear in mind when designing the insulation of transformer terminals, and these will now be referred to briefly. The high-tension leads of a transformer may break down (i) by puncture of the insulation, or (2) by spark-over from terminal to case. If the transformer lead could be considered as an insulated cable with a suitable dielectric separating it from an outer concentric FIG. 23. Section through Insulated Conductor. metal tube of considerable length, the calculation of the puncture voltage (i) would be a simple matter. For instance, let r in Fig. 23 be the radius of the inner (cylindrical) conductor, and R the internal radius of the enclosing tube, the space between being filled with a dielectric of which the specific inductive capacity (k) is constant throughout the insulating material. The equipotential surfaces will be cylinders, and the 56 PRINCIPLES OF TRANSFORMER DESIGN flux density 'over the surface of any cylinder of radius ^ x and of length i cm., will be D = . 2TTX By Formula (10) the potential gradient is, D ^f In order to express this relation in terms of the total voltage E, it is necessary to substitute for the symbol ^ its equivalent ExC, and calculate the capacity C of the condenser formed by the rod and the con- centric tube. Considering a number of concentric shells in series, the elastance may be written as follows : i r C = J dx i R C = Substituting in (15), we have, jf G = jp volts per centimeter, . . (17) x log, the maximum value of which is at the surface of the inner conductor, where (18) This formula is of some value in determining the thick- ness of insulation necessary to avoid overstressing the INSULATION OF HIGH-PRESSURE TRANSFORMERS 57 dielectric; but it is not strictly applicable to trans- former bushings in which the outer metal surface (the bushing in the lid of the containing tank) is short in comparison with the diameter of the opening. The advantage of having a fairly large value for r is indicated by Formula (18), and a good arrangement is to use a hollow tube for the high-tension terminal, with the lead from the windings passing up through it to a clamping terminal at the top. Solid porcelain bushings with either smooth or cor- rugated surfaces may be used for any pressure up to 40,000 volts, but for higher pressures the oil-filled type or the " condenser " type of terminal is preferable. In designing plain porcelain bushings it is important to see that the potential gradient in the air space between the metal rod and the insulator is not liable to cause brush discharge, as this would lead to chemical action, and a green deposit of copper nitrate upon the rod. The calculations would be made as explained for the parallel- plate condensers in which a sheet of glass was inserted (see " Capacities in Series "), except that the elastances of the condensers are now expressed by Formula (16). 18. Oil-filled Bushings. The chief advantages of a hollow insulating shell filled with oil or insulating com- pound that can be poured in the liquid state, are the absence of air spaces where corona may occur, and the possibility of obtaining a more uniform and reliable insulation than with solid insulators such as porcelain, when the thickness is considerable. The metal ring by which such an insulator (see Fig. 24) is secured to the transformer cover usually takes the form of a cylinder 58 PRINCIPLES OF TRANSFORMER DESIGN of sufficient length to terminate below the surface of the oil. The advantage of this arrangement is that the dielectric flux over the surface of the lower part of the insulator is through oil only, and not as would otherwise be the case, through oil and air. With the two mate- rials of different dielectric constants, the stress at the surface of the oil may exceed the dielectric strength of air, in which case there would be corona or brush dis- charge which might practically short-circuit the air path and increase the stress over that portion of the surface which is under the oil. The bushing illustrated in Fig. 24 has been designed for a working pressure of 88,000 volts between high- tension terminal and case, the method of computation being, briefly, as follows: Applying the rule for sur- face leakage distances previously given, this dimension is found to be 0.5 +- = 44. 5 in. The insulator need not, however, measure 44.5 in. in height above the cover of the transformer case, because corrugations can be used to obtain the required length. A safe rule to follow in deciding upon a minimum height, i.e., the direct distance in air between the terminal and the grounded metal, is to make this dimension at least as great as the distance between needle points that would just withstand the test voltage without sparking over. The test pressure is usually twice the working pressure plus 1000 volts, or 177 kv. (r.m.s. value) in this par- ticular case. This value corresponds to a distance of about 48 cm., or (say) 19 in. In order that there may be an ample margin of safety, it will be advisable to make the total height of the insulator not less than 22 INSULATION OF HIGH-PRESSURE TRANSFORMERS 59 Porcelain Iron Sleeve carried below surface of oil Metal tube of 234 outside diaru ji Insulating tube around ifmetal cap and transformer I' lead H.T. Lead FIG. 24. Three-part Composition-filled Porcelain Transformer Bushing, Suitable for a Working Pressure of 88,000 Volts to Ground. 60 PRINCIPLES OF TRANSFORMER DESIGN in., apart from the number or depth of the corrugations. The actual height in Fig. 24 is 31 in. because the cor- rugations on the outside of the porcelain shell are neither very numerous nor very deep. In this connection it may be stated that a short insulator with deep corruga- tions designed to provide ample surface distance is not usually so effective as a tall insulator with either a smooth surface or shallow corrugations. The reason is that much of the dielectric flux from the high-tension terminal to the external sleeve or supporting framework passes through the flanges, the specific inductive capacity of which is two to three times that of the air between them. The result is an increased stress in the air spaces, which is equivalent to a reduction in the effective height of the insulator. In the design under consideration it is assumed that the hollow (porcelain) shell is filled with an insulating compound which is solid at normal temperatures, and that the joints therefore need not be so carefully made as when oil is used. The insulator consists of three parts only, which are jointed as indicated on the sketch. Oil-filled bushings for indoor use generally have a large number of parts, usually in the form of flanged rings with molded tongue-and-groove joints filled with a suitable cement. There is always the danger, however, that a vessel so constructed may not be quite oil- tight, therefore the solid compound has an advantage over the oil in this respect. The creepage distance over the surface of the insulator in oil may be very much less than in air. Applying the rule previously given, the minimum distance in this INSULATION OF HIGH-PRESSURE TRANSFORMERS 61 case would be 0.5 + (0.1 X88) =9.3 in. In the design illustrated by Fig. 24, however, this dimension has been increased about 50 per cent with a view to keeping the high-tension connections well away from the sur- face of the oil and grounded metal. To prevent the accumulation of conducting particles in the oil along the lines of stress, and afford increased protection with only a small addition in cost, it is advisable to slip one or more insulating tubes over the lower part of the ter- minal, as indicated by the dotted lines in the sketch. Corrugations on the surface of the insulator in the oil are usually unnecessary, and sometimes objectionable because they collect dirt which may reduce the effective creepage distance. Having decided upon the height and surface distances to avoid all danger of spark-over, the problem which remains to be dealt with is the provision of a proper thick- ness of insulation to prevent puncture. In order to avoid complication of the problem by considering the different dielectric constants (k) of the compound used for filling and of the external shell (assumed in this case to be porcelain), it may be assumed either that there is no difference in the dielectric constants of the two materials, or that the thickness of the inclosing shell of porcelain is negligibly small in relation to the total external diameter of the insulator. Either assump- tion, neglecting the error due to the limited length of the external metal sleeve,* permits the use of Formula (18), * The maximum stress in the dielectric might be 5 to 10 per cent greater than calculated by using formulas relating to very long cylin- ders. The corners at the ends of the outer cylinder should be rounded off to avoid concentration of dielectric flux at these places. 62 PRINCIPLES OF TRANSFORMER DESIGN giving the relation between the maximum potential gradient and the dimensions of the bushing, without correction. Suppose that the disruptive gradient of the insulating compound is 90 kv. per centimeter (maximum value) or 63.5 kv. per centimeter (r.m.s. value) of the alternating voltage. With a test pressure of 177 kv. and a margin of safety of 25 per cent, the value of E in Formula (18) will therefore be =177X1. 25X^2 =313 kv. Since the disadvantage of a very small value of r is evident from an inspection of the formula, the outside diameter of the inner tube is made 2.25 in. Then, since G= E r log e - logio- = - -^- = 1.216, r 2.54X1.125X90X2.303 whence ^ = 3.79, or (say) 3.75 in. An external diam- eter of 7.5 in. at the center of the insulator will there- fore be sufficient to prevent the stress at any point exceeding the rupturing value even under the test pres- sure. 19. Condenser Type of Bushing. If the total thick- ness of the insulation between the high-tension rod and the (grounded) supporting sleeve is divided into a num- ber of concentric layers by metallic cylinders, the con- centration of dielectric flux at certain points (leading to high values of the voltage gradient) is avoided. The bushing then consists of a number of plate condensers in series, with a definite potential difference between INSULATION OF HIGH-PRESSURE TRANSFORMERS 63 the plates. If the total radial depth of insulation is divided into a large number of concentric layers (of the same thickness), separated by cylinders of tinfoil (of the same area) , the several condensers would all have the same capacity. The dielectric flux density, and therefore the potential gradient, would then be the same in all the condensers, so that the outer layers of insula- tion would be stressed to the same extent as the inner layers, and the total radial depth of insulation would be less than when the stress distribution follows the logarithmic law (Formula 18) as in the case of the solid porcelain, or oil-filled, bushing. The section on the right-hand side of Fig. 25 is a diagrammatic representation of a condenser bushing shaped to comply with the assumed conditions of equal thicknesses of insulation and equal areas of the con- denser plates. With a sufficient number of concentric layers, the condition of equal potential difference be- tween plate and plate throughout the entire thickness would be approximated; but the creepage distance over the insulation between the edges of the metal cylinders would be much smaller for the outer layers than for layers nearer to the central rod or tube. It is equally, if not more, important to prevent excessive stress over the surface than in the body of the insulator, and a practical condenser type of terminal can be designed as a compromise between the two conflicting require- ments. By making the terminal conical in form, as indicated by the dotted lines on the right-hand side and the full lines on the left-hand side of the sketch (Fig. 25), neither of the ideal conditions will be exactly 64 PRINCIPLES OF TRANSFORMER DESIGN fulfilled, but practical terminals so constructed are easily manufactured, and give satisfaction on circuits up to Metal shield to control ^^distribution of dielectric field. ^H ^^^^p^^^^^ 100,000 Volts I .^ \ \ V ,Tinfoil ^Insulation 60,000 Volts .^^ 1 I , i 40,000 Volts '***" V \ Grounded \^ metal 20,000 ^ Zero potential ; \ f \ // ' al tube or rod, ling H.T. lead, Met forn FIG. 25. Illustrating Principle of Condenser Type Bushing. 150,000 volts. By varying the thickness of the indi- vidual insulating cylinders, it is an easy matter to design a condenser type terminal of which the con- INSULATION OF HIGH-PRESSURE TRANSFORMERS 65 densers in series all have the same capacity even while the outside surface is conical in shape as shown on the left-hand side of Fig. 25. This gives a uniform potential gradient along the surface, and results in a good practical form of condenser-type bushing. If the ends of the metal cylinders coincide with equi- potential surfaces having the same potential as that which they themselves attain by virtue of the respective capacities of the condensers in series, there will . be no corona or brush discharge at the edges of these cylin- ders. This ideal condition is represented diagram- matically in Fig. 25, where a large metal disk is shown at the top of the terminal. The object of this metal shield is to distribute the field between the terminal and the transformer cover in such a manner as to satisfy the above-mentioned condition. In practice, the ten- dency for corona to form at the exposed ends of the tin- foil cylinders is counteracted by treating the finished terminal with several coats of varnish, and surrounding it with an insulating cylinder filled with an insulating compound which can be poured in the liquid form and which solidifies at ordinary temperatures. This con- struction is shown in Fig. 26, which represents a prac- tical terminal of the condenser type. Compared with Fig. 24, it is longer, but appreciably smaller in diameter where it passes through the transformer cover. The dimensions of a condenser-type terminal such as illustrated in Fig. 26 may be determined approxi- mately as follows: Assuming the working pressure as 88,000 volts, and the maximum permissible potential gradient in the dielectric (usually consisting of tightly 66 PRINCIPLES OF TRANSFORMER DESIGN Metal dim to control flax distribution. e of Iniultttng mtril. Iniu tlng compound. FIG. 26. Condenser-type Transformer Bushing Suitable for a Working Pressure of 88,000 Volts. INSULATION OF HIGH-PRESSURE TRANSFORMERS 67 wound layers of specially treated paper) as 90 kv.,* the maximum radial thickness of insulation required . , total volts 313 xv will be - -=^ = 3.48 cm. or (say) 1.5 voltage gradient 90 in. to include an ample allowance for the dividing layers of metal foil. If the inner tube is 2.25 in. in diameter, as in the previous example, the external diameter over the insulation at the center will be 2.25 X3 = 5-25 in. instead of the 7.5 in. required for the previous design. It is customary to allow about 4000 volts per layer, and twenty-two layers of insulation alternating with twenty-two layers of tinfoil are used in this particular design. It is true that ideal conditions will not be actually fulfilled; the aggregate thickness of insulation might have to be slightly greater than 1.5 in., but the inner tube might be made 1.75 in. or 2 in. instead of 2.25 in., and a practical terminal for 88,000- volt service could undoubtedly be constructed with a diameter over the insulation not exceeding 5.25 in. The projection of the terminal above the grounded plate (the cover of the transformer case) need not be so great as would be indicated by the application of the practical rule previously given for surface leakage dis- tance, namely, that this distance should be(o.5H -- -1 \ 2 / in., where kv. stands for the working pressure. The reason why a somewhat shorter distance is permissible is that the surface of the terminal proper has been cov- ered by varnish and a solid compound, and so far as the enclosing cylinder is concerned, the stress along the sur- * Same as in the example of the compound-filled insulator. 68 PRINCIPLES OF TRANSFORMER DESIGN face of this cylinder will be fairly uniform, especially if a large flux-control shield is provided, as shown in Fig. 26. In order to avoid the formation of corona at the lower terminal (below the surface of the oil) this end may conveniently be in the form of a sphere, the diameter of which would depend upon the voltage and the prox- imity of grounded metal. The following particulars relate to a condenser type bushing actually in service on 80,000 volts. The layers of insulation are built up on a metal tube of 2.25 in. outside diameter. The diameter over the outside in- sulating cylinder is 5.3 in. This bushing has uniform capacity, the thickness of the inner and outer insulating wall being the same, namely 0.062 in. ; but the thickness of the intermediate cylinders is variable, the maximum being 0.073 in. for the twelfth and thirteenth cylinders. (A plot of the individual thickness forms a hyperbolic curve.) The static shield or " hat " is 9 in. diameter and 2 in. thick, the edge being rolled to a true semicircle. When provided with a casing filled with gum, and when the taper is such that the steps on the air end are 1.69 in. (total length = i.69X22=37.2 in.), there is no dif- ficulty in raising the voltage to 300,000 (r.m.s. value) without arc-over. The same bushing without a casing would arc-over at about 285,000 volts; but this can be raised to the same value as for the terminal with gum- filled casing if the size of the static shield is increased to about 2 "ft. diameter. When the arc-over voltage is reached, the discharge takes place between the edge of the v static shield and the flange which is bolted to the transformer case. CHAPTER III EFFICIENCY AND HEATING OF TRANSFORMERS 20. Losses in Core and Windings. The power loss in the iron of the magnetic circuit is due partly to hysteresis and partly to eddy currents. The loss due to hysteresis is given approximately by the formula Watts per pound = KnB 1 ' 6 /, where K h is the hysteresis constant which depends upon the magnetic qualities of the iron. The symbols B and/ stand, respectively, for the maximum value of the mag- netic flux density, and the frequency. An approximate expression for the loss due to eddy .currents is Watts per pound = K e (Bft) 2 , where / is the thickness of the laminations, and K e is a constant which is proportional to the electric conduc- tivity of the iron. With the aid of such formulas, the hysteresis and eddy current losses may be calculated separately, and then added together to give the total watts lost per pound of the core material; but it is more convenient to use curves such as those of Fig. 27, which should be plotted 70 PRINCIPLES OF TRANSFORMER DESIGN \ \ \ \ 2fil IgSi* s^sa o \ (9) EFFICIENCY AND HEATING OF TRANSFORMERS 71 from tests made on samples of the iron used in the con- struction of the transformer. These curves give the relation between maximum value of flux density, and total iron loss per pound at various frequencies. The curves of Fig. 27 are based on average values obtained with good samples of commercial transformer iron and silicon-steel; the thickness of the laminations being about 0.014 in. The cost of silicon-steel stampings is greater than that of ordinary transformer iron; but the smaller total iron loss resulting from the use of the former material will almost invariably lead to its adoption on economic grounds. The eddy-current losses are. smaller in the alloyed material than in iron laminations of the same thickness because of the higher electrical resistance of the former. The permeability of silicon-steel is slightly lower than that of ordinary iron, and this may lead to a somewhat larger magnetizing current; on the other hand, the modern alloyed transformer material (silicon-steel) is non-ageing, that is to say, it has not the disadvantage common to transformers constructed fifteen to twenty years ago, in which the iron losses increased appreciably during the first two or three years of operation. The " ageing " of the ordinary brands of transformer iron resulting in larger losses is caused by the material being maintained at a fairly high temperature for a consider- able length of time. The maximum flux density in transformer cores is generally kept below the knee of the B-H curve. As a guide for use in preliminary designs, usual values of B (gausses) are given below : 72 PRINCIPLES OF TRANSFORMER DESIGN APPROXIMATE VALUES OF B IN TRANSFORMER CORES /-as / = So or 60 Small lighting or distributing transformers: Ordinary iron Alloyed iron 8,000 to 11,000 10,000 to 13,000 5,000 to 7,000 8,000 to 1 1 ooo Power transformers: Ordinary iron 10,000 to 13,000 9,000 to 11,000 Alloyed iron 11,000 to 14,000 1 1, ooo to 14 ooo The losses in the iron core are usually less than one watt per pound, although they sometimes amount to 1.5 watts, and even 1.8 watts, per pound. The higher figures apply to large, artificially cooled, power trans- formers. Current Density in Windings. Even with well-ven- tilated coils (air blast), or improved methods of pro- ducing good oil circulation, the permissible current den- sity in the copper windings is limited by local heating. Jf the watts lost per pound of copper exceed a certain amount, there will be danger of internal temperatures sufficiently high to cause injury to the insulation. As a rough guide in deciding upon suitable values for trial dimensions in a preliminary design, the following approx- imate figures may be used : AVERAGE VALUES OF CURRENT DENSITY (A) IN COMMERCIAL TRANSFORMERS Type of Transformer. sSSTln?" Standard lighting transformers (oil -immersed; self-cooled). 800 to 1200 Transformers for use in Central Generating Stations, or Substations (oil-cooled, or air blast) noo to 1600 Large, carefully designed transformers, oil-insulated, with forced circulation of oil, or with water cooling-coils 1400 to 1900 EFFICIENCY AND HEATING OF TRANSFORMERS 73 When the current is very large, it is important to sub- divide the conductors to prevent excessive loss by eddy currents. When flat strips are used, the laminations must be in the direction of the leakage flux lines. It is advisable to add from 10 to 15 per cent to the calculated I 2 R loss when the currents to be carried are large, even after reasonable precautions have been taken to avoid large local currents by subdividing the conductors. The mere subdivision of a conductor of large cross- section does not always eliminate the injurious effects of local currents in the copper, because, unless each of the several conductors that are joined in parallel at the ter- minals does not enclose the same amount of leakage flux, there will be different e.m.f.'s developed in various sections of the subdivided conductor, and consequent lack of uniformity in the current distribution. This ob- jection can sometimes be overcome by giving the assem- bled conductor (of many parallel wires or strips) a half twist, and so changing the position of the individual conductors relatively to the leakage flux; but, in any case, once this cause of increased copper loss is recog- nized, it is generally possible to dispose and join together the several elements of a compound conductor so that the leakage flux shall affect them all equally. 21. Efficiency. The output of a single-phase trans- former, in watts, is W = EJ s cosd, where E s is the secondary terminal voltage; 7 S , -the sec- ondary current; and cos 0, the power factor of the secondary load. The percentage efficiency is then: W 100 X W + iron losses + copper losses 74 PRINCIPLES OF TRANSFORMER DESIGN All-day Efficiency. The all-day efficiency is a matter of importance in connection with distributing trans- formers, because, although the amount of the copper loss falls off rapidly as the load decreases, the iron loss continues usually during the twenty-four hours, and may be excessive in relation to the output when the trans- former is lightly loaded, or without any secondary load, during many hours in the day. What is understood by the all-day percentage efficiency is the ratio given below, the various items being cal- culated or estimated for a period of twenty-four hours: i ooX Secondary output in watt-hours Sec. watt-hrs.+watt-hrs. iron loss+watt-hrs. copper loss' It is in order that this quantity may be reasonably large that the iron losses in distributing transformers are usually less than in power transformers designed for the same maximum output. Efficiency of Modern Transformers. The alternating- current transformer is a very efficient piece of apparatus, as shown by the following figures which are an indication of what may be expected of well-designed transformers at the present time. FULL-LOAD EFFICIENCIES OF SMALL LIGHTING TRANS- FORMERS FOR USE ON CIRCUITS UP TO 2200 VOLTS Output, k.v.a. Efficiency (per cent) i From 94 . i to 96 2 From 94 . 6 to 96 . 5 5 From 95.5 to 97.3 10 From 96 . 4 to 97 . 9 20 From 97.2 to 98 . i 50 From 97.6 to 98 . 4 EFFICIENCY AND HEATING OF TRANSFORMERS 75 For a given cost of materials, the efficiency will improve with the higher frequencies, and a transformer designed for a frequency of 25 would rarely have an efficiency higher than the lower limit given in the above table, while the higher figures apply mainly to transformers for use on 6o-cycle circuits. The highest efficiency of a lighting transformer usually occurs at about three-quarters of full load. Typical figures for a 5 k.v.a. lighting transformer for use on a 5o-cycle circuit are given below. Core loss = 46 watts. Copper loss (full load) = 114 watts. Calculated efficiency (100 per cent power factor) : At full load, 0.969. At three-quarters full load, 0.9713. At one-half full load, 0.9707. At one-quarter full load, 0.9583. FULL-LOAD EFFICIENCIES OF POWER TRANSFORMERS FOR USE ON 66,ooo-voLT CIRCUITS (100 per cent power factor) Output, k.v.a. Efficiency, per cent. 400 From 97.3 to 97 . 8 800 From 97 . 7 to 98 . 2 1 200 From 97.9 to 98 . 4 2000 From 98 . i to 98 . 7 2600 From 98 . 2 to 98 . 8 The manner in which the efficiency of large power transformers falls off with increase of voltage (involving loss of space taken up by insulation) is indicated by the 76 PRINCIPLES OF TRANSFORMER DESIGN following figures, which refer to 1000 k.v.a. single-phase units designed for use on 5o-cycle circuits. H.T. Voltage. Fu " Load Effic ^ ncy (Approximate) Per cent. 22,OOO 98.8 33> 000 ---- ' 98.7 44,000 98 . 5 66,000 . 98.3 88,000 , 98.0 110,000 97.8 The figures given below are actual test data showing the performance of some single-phase, oil-insulated, self- cooling, power transformers recently installed in a hydro- electric generating station in Canada: Output 400 k.v.a. Frequency /=6o Primary volts 2,200 Secondary volts 22,000 Core loss 1^760 watts Full-load copper loss 3, 550 watts Exciting current, 2.15 per cent, of full-load current. Temperature rise (by thermometer) after contin- uous full-load run, 36 C. Efficiency on unky power factor load : At 1.25 times full load.. . . 98.57 per cent At full load 98 . 7 At three-quarter s full load . 98 . 7 5 At one-half full load 98 . 65 At one-quarter full load. . . 98 . o EFFICIENCY AND HEATING OF TRANSFORMERS 77 It should be stated that the core loss in these trans- formers was exceptionally low, being only 0.44 per cent of the k.v.a. output. The core losses in modern trans- formers will usually lie between the limits stated below: K.v.a. Output. Volts. Percentage Core Loss 100 Xcore loss, watts. rated volt-ampere output' 500. ... | 22,000 66,000 0.75 to 0.95 I . tO 1.2 r 22,000 o . 6 to o . 7 IOOO. j 66,000 0.7 to i . o I IIO,OOO 0.8 to 1. 15 r 22,000 0.5 to 0.65 2000 J. 66,000 0.55 to 0.7 ( 110,000 0.7 to 0.95 4000 | 66,000 110,000 0.5 to o . 6 0.6 to o. 75 The core losses in small transformers for use on lighting circuits up to 2200 volts are usually less than i per cent for all sizes above 3 kw. They may be as low as 0.5 per cent in a 50 kw. distributing transformer, and as high as 2.5 per cent in a i kw. transformer. The frequency, whether 25 or 60, does not greatly influence the customary allowance for core loss. Efficiency when Power Factor oj Load is Less than Unity. The total full-load losses (iron -f- copper) may be ex- pressed as a percentage of the k.v.a. output. Assume that these losses are equal to a(k.v.a.). Then at any power factor, cos 0. 78 PRINCIPLES OF TRANSFORMER DESIGN Efficiency = - (k.v.a.) cos 0+a(k.v.a.) cos cos 0+a Let T\ stand for the efficiency at unity power factor, then and whence the efficiency at any power factor, cos 8, is cos 6 cos y ?7 / As an example, calculate the full-load efficiency of a transformer on a load of 0.75 power factor, given that the efficiency on unity power factor is 0.969. The ratio of the total losses to the k.v.a. output is 10.069 d= 2 ^ = 0.969 whence the efficiency at 0.75 power factor is EFFICIENCY AND HEATING OF TRANSFORMERS 79 22. Temperature of Transformer Windings. Insu- lating materials such as cotton and paper, specially treated with insulating compounds or immersed in oil, may be subjected to a temperature up to, but not exceeding 105 C. The hottest spot of the winding cannot be reached by a thermometer, and it is therefore customary to add 15 C. to the temperature registered by a thermometer placed at the hottest accessible part of a transformer under test. The room temperature is frequently as high as 35 C. and the maximum permis- sible rise in temperature above that of -the surrounding air may be arrived at as follows : Permissible hottest spot temperature. 105 Hottest spot correction 15 Difference 90 Assumed room temperature 35 Difference ( = permissible temperature rise) 55 Thus, under the worst conditions of heating, the per- missible temperature rise should not exceed 55 C. when the measurements are made with a thermometer. A more reliable means of arriving at transformer tempera- tures is to calculate these from resistance .measurements of the windings. Such measurements usually give some- what higher temperatures than when thermometers are used, and a hottest spot correction of 10 C. is then gen- erally recognized as sufficient. It should be noted, 80 PRINCIPLES OF TRANSFORMER DESIGN however, that room temperatures of 40 C. are not impossible, and it is therefore customary to limit the observed rise in temperature to 55 C. even when the resistance method of measuring temperatures is adopted. Transformers are usually designed to withstand an overload of two hours' duration after having been in continuous operation under normal full-load conditions. Either of the following methods of rating is to be found in modern transformer specifications: (1) The temperature rise not to exceed 40 C. on con- tinuous operation at normal load, and 55 C. after an additional two hours' run on 25 per cent overload. (2) The temperature rise not to exceed 35 C. on con- tinuous operation at normal load, and 55 C. after an additional two hours' run on 50 per cent overload. On account of the slow heating of the iron core, large oil-cooled transformers may require ten, or even twelve hours to attain the final temperature. 23. Heat Conductivity of Insulating Materials. Be- fore discussing the means by which the heat is carried away from the external surface of the coils, it will be advisable to consider how the designer may predetermine approximately the difference in temperature between the hottest spot and the external surface of the windings. Calculations of internal temperatures cannot be made very accurately; but the nature of the problem is indi- cated by the following considerations: Fig. 28 is supposed to represent a section through a very large flat plate, of thickness t, consisting of any homogeneous material. Assume a difference of tem- perature of T d = (T-To)C. to be maintained between EFFICIENCY AND HEATING OF TRANSFORMERS 81 the two sides of the plate, and calculate the heat flow (expressed in watts) through a portion of the plate of area wXl. The resistance offered by the material of the plate to the passage of heat may be expressed in thermal ohms, the thermal ohm being denned as the thermal resistance which causes a drop of i C. per watt I Watts =W FIG. 28. Diagram Illustrating Heat Flow through Flat Plate. of heat flow; or, if R h is the thermal resistance of the heat path under consideration, T T" *- (I (19) which permits of heat conduction problems being solved by methods of calculation similar to those used in con- nection with the electric circuit. 82 PRINCIPLES OF TRANSFORMER DESIGN Let k be the heat conductivity of the material, ex- pressed in watts per inch cube per degree Centigrade difference of temperature between opposite sides of the Y FIG. 29. Heat Conductivity: Heat Generated Inside Plate. cube, then the watts of heat flow crossing the area (wXl) square inches, as indicated in Fig. 28 is \ Fig. 29 illustrates a similar case, but the heat is now supposed to be generated in the mass of the material itself. We shall still consider the plate to be very large EFFICIENCY AND HEATING OF TRANSFORMERS 83 relatively to the thickness, so that the heat flow from the center outward will be in the direction of the hori- zontal dotted lines. A uniformly distributed electric current of density A amperes per square inch is supposed to be flowing to or from the observer, and the highest temperature will be on the plane YY' passing through the center of the plate. Assuming this plate to be of copper with a resistivity of o.84Xio~ 6 ohms per inch cube at a temperature of about 80 C., the watts lost in a section of area (xXw) sq. in. and length / in, will be W x = (Axw) 2 X 0.84X10- X xw (21) By adapting Formula (20) to this particular case, the difference of temperature between the two sides of a section dx in. thick is seen to be dTd =w x x dx whence, 0.84 X A 2 A" , T<L = r , I xdx I O /v / 0.84A 2 / 2 , = ^7-777 degrees Centigrade. . The value of k for copper is about 10 watts per inch cube per degree Centigrade. The problem of applying these principles to the prac- tical case of a transformer coil is complicated by the fact 84 PRINCIPLES OF TRANSFORMER DESIGN that the heat does not travel along parallel paths as in the preceding examples, and, further, that the thermal conductivity of the built-up coil depends upon the rel- ative thickness of copper and insulating materials, a relation which is usually different across the layers of winding from what it is in a direction parallel to the layers. A P, ,,. > m ,,,,,,,,,A r !**(B o ta ^ >- 5 J^J 1 r>- ?>- *'"L"y<' f y f y f yr< t f ^|f rr^' u FIG. 30. Diagram Illustrating Heat Paths in a Transformer Coil of Rectangular Cross-section. Fig. 30 represents a section through a transformer coil wound with layers of wire in the direction CM; the number of layers being such as to produce a total depth of winding equal to twice OB. The whole of the outside surface of this coil is supposed to be maintained at a constant temperature by the surrounding oil or air. In other words, it is assumed that there is a constant difference of temperature of T^ degrees between the hottest spot (supposed to be at the center 0) and any point on the surface of the coil. EFFICIENCY AND HEATING OF TRANSFORMERS 85 The heat generated in the mass of material is thought of as traveling outward through the walls of successive imaginary spaces of rectangular section and length / (measured perpendicularly to the plane of the section shown in Fig. 30), as indicated in the figure, where CDEF is the boundary of one of these imaginary spaces, the walls of which have a thickness dx in the direction OA , and a thickness dx ( } in the direction OB. (OB\ . (OA) " According to Formula (19), we can say that the dif- ference of temperature between the inner and outer boundaries of this imaginary wall is d 7^ = heat loss, in watts, occurring in the space CDEFXthe thermal resistance of the boundary walls. It is proposed to consider the heat flow through the portion of the boundary surface of which the area is CDEF XL If W x stands for the watts passing through this area, we can write 2DElkg { 2CDlk h dx dx which simplifies into 'OB , , ' ' (23) 86 PRINCIPLES OF TRANSFORMER DESIGN In order to calculate W x it is necessary to know not only the current density, A, but also the space factor, or ratio of copper cross-section to total cross- section. Let a stand for the thickness of copper per inch of total thickness of coil measured in the direction OA ; and let b stand- for a similar quantity measured in the direction OB ; the space factor is then (aXb), and 0.84X10-6- Inserting this value of W x in (23), and making the necessary simplifications, we get dT d = 10' whence, by integration between the limits x a and x=OA, T d = r~ /TTTTTT de S- Cent - ( 2 4) 2X'lC Except for the obvious correction due to the intro- duction of the space factor (ab), the only difference between this formula and Formula (22) is that the thermal conductivity, instead of being k a , as it would be if the heat flow were in the direction OA only, is EFFICIENCY AND HEATING OF TRANSFORMERS 87 replaced by the quantity in brackets in the denominator of Formula (24). This quantity may be thought of as a fictitious thermal conductivity in the direction OA, which, being greater than k aj provides the necessary cor- rection due to the fact that heat is being conducted away in the direction OB, thus reducing the difference of tem- perature between the points and A. Calculation of k a and kb. Let k c and ki, respectively, stand for the thermal con- ductivity of copper and insulating materials as used in transformer construction. The numerical values of these quantities, expressed in watts per inch cube per degree Centigrade, are k c = 10 and k t = 0.0033. It follows that 10 ( , (25) <z_ (i a) a-\- 3000(1 a) and similarly, 10 6+3000(1-6) where a and b are the thickness of copper per inch of coil in the directions OA and OB, respectively, as pre- viously defined. Example. Suppose a transformer coil to be wound with 0.25X0.25 in. square copper wire insulated with cotton o.oi in. thick, and provided with extra insulation of 0.008 in. fullerboard between layers. There are twelve layers of wire and seven wires per layer. Assume the current density to be 1400 amperes per square inch, -- r~ , - ^rj 2 XI06 [0.0448 +0.033 2 \ 88 PRINCIPLES OF TRANSFORMER DESIGN and calculate the hottest spot temperature if the outside surface of the coil is maintained at 75 C. 0.25 0.25 a = -. = 0.926; b = -5 = 0.9; whence space factor 0.27 0.278 (ab) =0.833. By Formulas (25) and (26), ^ = 0.0448, and ^ = 0.033 2; OA =3.5X0.27=0.945, and 0^ = 6X0.278 = 1.67 in. By Formula (24), =n Cent., and the hottest spot temperature = 75 + 11 =86. C. 24. Cooling Transformers by Air Blast. Before the advantages of oil insulation had been realized, trans- formers were frequently enclosed in watertight cases, the metal of these cases being separated from the hot parts of the transformer by a layer of still air. This resulted either in high temperatures or in small kilowatts output per pound of material. Air insulation is still used in some designs of large transformers for pressures up to about 33,000 volts; but efficient cooling is ob- tained by forcing the air around the windings and through ducts provided not only between the coils, but also between the coils and core, and between sec- tions of the core itself. Since all the heat losses which are not radiated from the surface of the transformer case must be carried away by the air blast, it is a simple matter to calculate the EFFICIENCY AND HEATING OF TRANSFORMERS 89 weight (or volume) of air required to carry away these losses with a given average increase in temperature of outgoing over ingoing air. A cubic foot of air per minute, at ordinary atmospheric pressures, will carry away heat at the rate of about 0.6 watt for every degree Centigrade increase of tempera- ture. Thus, if the difference of temperature between outgoing and ingoing air is 10 C., the quantity of air which must pass through the transformer for every kilo- watt of total loss that is not radiated from the surface of the case, is = = 1 66 cu. ft. per minute. v 0.6X10 If the average increase in temperature of the air is from 10 to 15 C., the actual surface temperature rise of the windings may be from 40 to 50 C. ; the exact figure being difficult to calculate since it will depend upon the size and arrangement of the air ducts. The temperature of the coils is influenced not only by the velocity of the air over the heated surfaces, but also by the amount of the total air supply which comes into intimate contact with these surfaces. With air passages about J in. wide, and an average air velocity through the ducts ranging from 300 to 600 ft. per minute, the temperature rise of the coil surfaces will usually be from four to eight times the rise in temperature of the circulating air. Thus, although it is not possible to predetermine the exact quantity of air necessary to maintain the 90 PRINCIPLES OF TRANSFORMER DESIGN transformer windings at a safe temperature, this may be expressed approximately as: Cubic feet of air per minute r o /-i , Wt W r for 50 C. temperature rise = - --, r M r o.oX-%- of coil surface where W t total watts lost in transformer ; and W r = portion of total loss dissipated from surface of tank. The latter quantity may be estimated by assuming the temperature of the case to be about 10 C. higher than that of the surrounding air, and calculating the watts radiated from the case with the aid of the data in the succeeding article. Assuming W r to be 25 per cent of Wt, the Formula (27) indicates that about 150 cu. ft. of air per minute per kilowatt of total losses would be necessary to limit the temperature rise of the coils to 50 C. With poorly designed transformers, and also in the case of small units, the amount of air required may be appreciably greater. It is true that, in turbo-generators, an allowance of 100 cu. ft. per minute per kilowatt of total losses, is gen- erally sufficient to limit the temperature rise to about 50 C.; but, owing to the churning of the air due to the rotation of the rotor, it would seem that the necessary supply of air is smaller for turbo-generators than for transformers. EFFICIENCY AND HEATING OF TRANSFORMERS 91 Filtered air is necessary in connection with air-blast cooling; otherwise the ventilating ducts are liable to become choked up with dirt, and high temperatures will result. Wet air niters are very satisfactory and desir- able, provided the amount of moisture in the air passing through the transformers is not sufficient to cause a deposit of water particles on the coils. Air containing from i to 3 per cent of free water in suspension is a much more effective cooling medium than dry air. It would probably be inadvisable to use anything but dry air in contact with extra-high voltage apparatus; but trans- formers for very high pressures are not designed for air-blast cooling.* 25. Oil-immersed Transformers Self Cooling. The natural circulation of the oil as it rises from the heated surfaces of the core and windings, and flows do w a ward near the sides of the containing tank, wil-l lead to a tem- perature distribution generally as indicated in Fig. 31. The temperature of the oil at the hottest part (close to the windings at the top of the transformer) will be some- what higher than the maximum temperature of the tank, which, however, will be hotter in the neighborhood of the oil level than at other parts of its surface. The average temperature of the cooling surface in contact with the air bears some relation to the highest oil temperature, and, since this relation does not vary greatly with different designs of transformer, or case, a curve such * Some useful data on the relative cooling effects of moist and dry air, together with test figures relating to a i2-kw. air-cooled transformer, will be found in Mr. F. J. Teago's paper " Experiments on Air-blast Cooling of Transformers," in the Jour. Inst. E. ., May i, 1914 ,Vol. 52, page 563. 92 PRINCIPLES OF TRANSFORMER DESIGN as Fig. 32 may be used for calculating the approximate tank are a necessary to prevent excessive oil temperatures. The oil temperature referred to in Fig. 32 is the dif- ference in degrees Centigrade between the temperature of the hottest part of the oil and the air outside the tank. Temperature of < FIG. 31. Distribution of Temperature with Transformer Immersed in Oil. This will be somewhat greater than the temperature rise of any portion of the transformer case; but the curve indicates the (approximate) number of watts that can be dissipated by radiation and air currents per square inch of tank surface. The curve is based on average figures obtained from tests on tanks with smooth surfaces EFFICIENCY AND HEATING OF TRANSFORMERS 93 54 50 46 42 38 34 30 o 26 18 14 10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 '4= > Watt8 dissipated per sq. in. of tank surface FIG. 32. Curve for Calculating Cooling Area of Transformer Tanks. 94 PRINCIPLES OF TRANSFORMER DESIGN (not corrugated), the surface considered being the total area of the (vertical) sides plus one-half the area of the lid. The cooling effect of the bottom of the tank 'is practically negligible, and is not to be included in the calculations. Example. What will be the probable maximum tem- perature rise of the oil in a self-cooling transformer with a total loss of 1200 watts, the tank of sheet-iron with- out corrugations measuring 2 ft.X2 ft.X3-5 ft. high? The surface for use in the calculations is S = (3. 5X8) +2 = 30 sq. ft., whence 1 200 w = =0.278, 30X144 which', according to Fig. 32, indicates a 43 C. rise of temperature for the oil. The temperature of the windings at the hottest part of the surface in contact with the oil might be from 5 to 10 C. higher than the maximum oil temperature as meas- ured by thermometer. Assume this to be 7 C. Assume also that the room temperature is 35 C. and that the difference of temperature (To) between the coil surface and the hottest spot of the windings as calculated by the method explained in Art. 23 is 13 C. Then the hottest spot temperature in the transformer under con- sideration would be about 35+43 + 7 + 13 = 98 C. 26. Effect of Corrugations in Vertical Sides of Con- taining Tank. The cooling surface in contact with the air may be increased by using corrugated sheet-iron tanks in place of tanks with smooth sides. It must not, EFFICIENCY AND HEATING OF TRANSFORMERS 95 however, be supposed that the temperature reduction will be proportional to the increase of tank surface provided in this manner; the watts radiated per square inch of surface of a tank with corrugated sides will always be appreciably less than when the tank has smooth sides. Not only is the surface near the bottom of the corruga- tions less effective in radiating heat than the outside portions; but the depth and pitch of the corrugations will affect the (downward) rate of flow of the oil on the inside of the tank, and the (upward) convection cur- rents of air on the outside. It is practically impossible to develop formulas which will take accurate account of all the factors involved, and recourse must therefore be had to empirical formulas based on available test data together with such reason- able assumptions as may be necessary to render them suitable for general application. If X is the pitch of the corrugations, measured on the outside of the tank, and / is the surface width of material per pitch (see the sketch in Fig. 33), the ratio of the actual tank surface to the surface of a tank without corrugations is -. The heat dissipation will not be X in this proportion because, although the cooling effect will increase as / is made larger relatively to X, the additional surface becomes less and less effective in radiating heat as the depth of the corrugations increases without a corresponding increase in the pitch. It is convenient to think of the surface of an equivalent smooth tank which will give the same temperature rise of the oil as will be obtained with the actual tank. 96 PRINCIPLES OF TRANSFORMER DESIGN If we apply a correction to the actual pitch, X, and obtain an equivalent pitch, \ e , the ratio k = A is a factor by which the tank surface (neglecting corru- 2.6 2.4 2.2 1.G 1.4 1.2 0.2 0.3 0.4 0.5 0.7 0.8 0.9 FIG. 33. Curve Giving Factor k for Calculating Equivalent Cooling Surface of Tanks with Corrugated Sides. gations) must be multiplied in order to obtain the equivalent or effective surface. If all portions of the added surface were equally effective in radiating heat, no correcting factor would be required, and the equiva- lent pitch would be obtained by adding to X the quan- EFFICIENCY AND HEATING OF TRANSFORMERS 97 tity (/ X); but since a modifying factor is needed, the writer proposes the formula , .... (28) wherein the additional surface provided by the cor- 2\ rugations is reduced in the ratio - which becomes /~T~X unity when / = X. A modifying factor of this form not only seems reasonable on theoretical grounds, but it is required in an empirical formula based on available experimental data. It follows that 't x or, if- Values of &, as obtained from this formula for different values of n, may be read off the curve Fig. 33. Example. What would have been the temperature rise of the oil if, instead of the smooth-side tank of the preceding example (Art. 25), a tank of the same external dimensions had been provided with corrugations 2 in. deep, spaced ij in. apart? The approximate value of / is 1.25+4 = 5.25 in. Whence w = ? -=0.238; and from the curve, Fig. 30, = 2.23. / 98 PRINCIPLES OF TRANSFORMER DESIGN The equivalent tank surface is 5 = (3.5X8X2. 23) + 2 = 64.5 sq. in., whence 1200 w = - = 0.129, 64.5X144 which, according to Fig. 32, indicates a 27 C. rise of temperature, as compared with 43 C. with the smooth- surface tank of the same outside dimensions. 27. Effect of Overloads on Transformer Tempera- tures. Since the curve of Fig. 32 is not a straight line, it follows that the watts dissipated per square inch of tank surface are not directly proportional to the differ- ence between the oil, and room, temperatures. The approximate relation, according to this curve, is Temperature rise = constant X w' 6 , . . (31) which may be used for calculating the temperature rise of a self-cooling oil-immersed transformer when the tem- perature rise under given conditions of loading is known. Example. Given the following particulars relating to a transformer'. Core loss = 100 watts, Copper loss (full load) = 200, Final temp, rise (full load) of the 011 = 35 C. Calculate the final temperature rise after a continuous run at 20 per cent overload. For an increase of 20 per cent in the load, the copper EFFICIENCY AND HEATING OF TRANSFORMERS 99 loss is 2ooX (i. 2)2 = 288 watts; whence, according to Formula (31): - 6 /2SS-hIOoV' u o r^ Temperature rise = 35X1- - I =41 C. approx. \2OO+IOO/ The calculation of temperature rise resulting from an overload of short duration is not so simple. It is neces- sary to take account of the specific heat of the materials, especially the oil, because the heat units absorbed by the materials have not to be radiated from the tank surface, and the calculated temperature rise would be too high if this item were neglected. The specific heat of a substance is the number of calor- ies required to raise the temperature of i gram i C., the specific heat of water being taken as unity. The specific heat of copper is 0.093, an d for an average quality of transformer oil, it is 0.32. One gram-calorie (i.e., the heat necessary to raise the temperature of i gram of water i C.)=4.i83 joules (or watt-seconds). Also, i lb.=453.6 grams. It follows that the amount of energy in watt-seconds necessary to raise M c pounds of copper T C. is Watts X time in seconds = 4. 183X0.093X453. 6 M C T = 177 M C T (for copper). Similarly, if we put M for the weight of oil, in pounds, and replace the figure 0.093 by 0.32, we get Watts X time in seconds = 6 10 M T (for oil). 100 PRINCIPLES OF TRANSFORMER DESIGN In the case of an overload after the transformer has been operating a considerable length of time on normal full load, all the additional losses occur in the copper coils, and it is generally permissible to neglect the heat absorption by the iron core. We shall, therefore, assume that the additional heat units which are not absorbed by the copper pass into the oil, and that the balance, which is not needed to heat up the oil, must be dissipated by radiation and convection from the sides of the containing tank. It will greatly simplify the cal- culations if we further assume that the watts dissipated per square, inch of tank surface per degree difference of temperature are constant over the range of temperature involved in the problem. (By estimating the average temperature rise, and finding w on the curve, Fig. 32, w a suitable value for the quantity may be selected.) If W t = total watts lost (iron -[-copper), the total energy loss in the interval of time dt second is Wtdt. If the increase of temperature during this interval of time is dx degree Centigrade, the heat units absorbed by the copper coils and the oil are K s dx, where K s = The difference between these two quantities represents the number of joules, or watt-seconds, of energy to be * In order to simplify the calculations, it has been (incorrectly) assumed that the temperature rise of the copper is the same as that of the oil. It will, of course, be somewhat greater; but since the heat absorbed by the copper is small compared with that absorbed by the oil, this assump- tion will not lead to an error of appreciable magnitude. EFFICIENCY AND HEATING OF 'JRANSSI-ORMERS iCl radiated from the tank surface during the interval of time dt second; whence, , .... (32) where Kr = t&nk surface in square inches X radiation coefficient, in watts per square inch per i C. rise, and # = the initial oil temperature rise (which has been increased, by the amount dx) . Equation (32) may be put in the form dL = Ks dx W t The limits for x are the initial oil temperature TQ and the final oil temperature T t , which is reached at the end of the time /. Therefore, (33) If time is expressed in minutes, and common logs, are used, we have, w, T \ K s , lK~r~ 10 \ . , . - minutes. . . (34) \ Wt T I \K- Tl / 102 PRINCIPLES OF TRANSFORMER DESIGN In order to facilitate the use of this formula, the meaning of the symbols is repeated below: Wt total watts lost (iron + copper) ; K T =SX radiation coefficient expressed in watts per square inch per i C. rise of temperature of the oil; where S = tank surface in square inches, as de- fined in Art. 25, corrected if necessary for corrugations (Art. 26). where M c = weight of copper (pounds) ; and MQ = weight of oil (pounds) ; TO = initial temperature of oil (degrees C.); T t = temperature of oil (degrees C.) after the overload (producing the total losses Wt) has been on for t m minutes. Example. Using the data of the preceding example, the full-load conditions are: Core loss = 100 watts; Copper loss = 200 watts; Temperature rise = 3 5 C. Referring to Fig. 29, the value for w for a temper- ature rise of 35 is 0.193, from which it follows that the effective tank surface is S = ~ = 1550 sq. in. * yo Given the additional data: Weight of copper = 65 lb., Weight of oil = 140 lb. EFFICIENCY AND HEATING OF TRANSFORMERS 103 calculate the time required to raise the oil from T = 35 C. to r* = 45 C. on an overload of 50 per cent. The copper loss is now 200 X (i-5) 2 =450 watts, whence Wt 100+450 = 550 watts. The cooling coefficient (from curve, Fig. 32), for an average temperature rise of - -=40 C., is - = 0.00606, whence, K r = # s =(l77X65) + (610X140) =97,000; and, by Formula (34), 94 28. Self-cooling Transformers for Large Outputs. The best way to cool large transformers is to provide them with pipe coils through which cold water is circu- lated, or, alternatively, to force the oil through the ducts and provide means for cooling the circulating oil outside the transformer case. When such methods cannot be adopted as in most outdoor installations and other sub-stations without the necessary machinery and at- tendants the heat from self-cooling transformers of large size is dissipated by providing additional cooling surface in the form of tubes, or flat tanks of small volume 104 PRINCIPLES OF TRANSFORMER DESIGN and large external surface, connected to the outside of a central containing tank. Unless test data are available in connection with the particular design adopted, judg- ment is needed to determine the effective cooling surface (see Art. 26) in order that the curve of Fig. 32 or such cooling data as may be available for smooth-surface tanks may be used for calculating the probable tem- perature rise. In the tubular type of transformer tank which is pro- vided with external vertical tubes connecting the bottom of the tank to the level, near the oil surface, where the temperature is highest (as roughly illustrated by Fig. 34), the tubes should be of fairly large diameter with sufficient distance be- tween them to allow free circulation of the air and efficient radiation. It is not economical to use a very large number of small tubes closely spaced with a view to obtaining a large cooling surface, because the FIG. 34. Transformer Case with r . , . , , Tubes to Provide Additional Cool- CXtra SUrfaCC obtained by ing Surface. such means is not as effective as when wider spacing is used. If the added pipe surface, A p , is 1.5 times the tank surface, A tj without the pipes, the effective cooling surface will be about S = (A t +A p )Xo.g] but, EFFICIENCY AND HEATING OF TRANSFORMERS 105 with a greatly increased surface obtained by reducing the spacing between the pipes, the correction factor might be very much smaller than 0.9. 29. Water-cooled Transformers. The cooling coil should be constructed preferably of seamless copper tube about ij in. diameter, placed near the top of the tank, but below the surface of the oil. If water is passed through the coil, heat will be carried away at the rate of 1000 watts for every 3! gals, flowing per minute when the difference of temperature between the outgoing and ingoing water is i C. Allowing 0.25 gal. per minute, per kilowatt, the average temperature rise of the water will be = 15 C. The temperature rise of the oil is considerably greater than this: it will depend upon the area of the coil in contact with the oil and the condition of the inside surface, which may become coated with scale. An allowance of i sq. in. of coil surface per watt is customary; but the rate at which heat is transferred from the oil to the water may be from 2 to 2j times as great when the pipes are new than after they have become coated with scale. It may, therefore, be neces- sary to clean them out with acid at regular intervals, if the danger of high oil temperatures is to be avoided. Example. Calculate the coil surface and the quan- tity of water required for a transformer with total losses amounting to 6 kw., of which it is estimated that 2 kw. will be dissipated from the outside of the tank. Surface of cooling coil = 6000 2000 = 4000 sq. in. Assuming a diameter of ij in., the length of tube in 4.000 the coil will have to be -7 = 85 ft. - 106 PRINCIPLES OF TRANSFORMER DESIGN The approximate quantity of water required will be 0.25X4 = 1 gal. per minute. 30. Transformers Cooled by Forced Oil Circulation. The transformer and case are specially designed so that the oil may be forced (by means of an external pump) through the spaces provided between the coils and be- tween the sections of the iron core. The ducts may be narrower than when the cooling is by natural circula- tion of the oil. The capacity of the oil pump may be estimated by allowing a rate of flow of oil through the ducts ranging from 20 to 30 ft. per minute. It is not essential that the oil be cooled outside the transformer case; in some modern transformers, the con- taining tank proper is surrounded by an outer case, and the space between these two shells contains the cooling coils through which water is circulated. These coils, instead of being confined to the upper portion of the transformer case, as when water cooling is used without forced oil circulation, may occupy the whole of the space between the inner and outer shells of the containing tank. The oil circulation is obtained by forcing the oil up through the inner chamber and downward in the space surrounding the water cooling-coils. Such systems of artificial circulation of both oil and water are very effective in connection with units of large output; but they could not be applied economically to medium-sized or small units. CHAPTER IV MAGNETIC LEAKAGE IN TRANSFORMERS REACTANCE- REGULATION 31. Magnetic Leakage. Assuming the voltage applied to the terminals of a transformer to remain constant, it follows that the flux linkages necessary to produce the required back e.m.f. can readily be calculated. The (vectorial) difference between the applied volts and the induced volts must always be exactly equal to the ohmic drop of pressure in the primary winding. Thus, the total primary flux linkages (which may include leakage lines) must be such as to induce a back e.m.f. very nearly equal to the applied e.m.f. the primary IR drop being comparatively small. When the secondary is open-circuited, practically all the flux linking with the primary turns links also with the secondary turns; but when the transformer is loaded, the m.m.f. due to the current in the secondary winding has a tendency to modify the flux distribution, the action being briefly as follows: The magnetomotive force due to a current I s flowing in the secondary coils would have an immediate effect on the flux in the iron core if it were not for the fact that the slightest tendency to change the number of flux lines through the primary coils instantly causes the primary current to rise to a value I p such that the resultant 107 108 PRINCIPLES OF TRANSFORMER DESIGN ampere turns (I P T P I S T S ) will produce the exact amount of flux required to develop the necessary back e.m.f. in the primary winding. Thus, the total amount of flux linking with the primary turns will not change appreciably when current is drawn from the secondary terminals; but the secondary m.m.f. together with an exactly ^qual but opposite primary magnetizing effect will cause some of the flux which previously passed through the secondary core to " spill over " and avoid some, or all, of the secondary turns. This reduces the secondary volts by an amount exceeding what can be accounted for by the ohmic resistance of the windings. Although it is possible to think of a leakage field set up by the secondary ampere turns independently of that set up by the primary ampere turns, these imaginary flux components must be superimposed on the main flux common to both primary and secondary in order that the resultant magnetic flux distribution under load may be realized. The leakage flux is caused by trie com- bined action of primary and secondary ampere turns, and it is incorrect, and sometimes misleading, to think of the secondary leakage reactance of a transformer as if it were distinct from primary reactance, and due to a particular set of flux lines created by the secondary current. In order to obtain a physical conception of magnetic leakage in transformers it is much better to assume that the secondary of an ordinary transformer has no ^//-inductance, and that the loss of pressure (other than IR drop) which occurs under load is caused by the secondary ampere turns diverting a certain amount of magnetic flux which, although it still links with the MAGNETIC LEAKAGE IN TRANSFORMERS 109 primary turns, now follows certain leakage paths instead of passing through the core under the secondary coils. 32. Effect of Magnetic Leakage on Voltage Regula- tion. The regulation of a transformer may be defined as the percentage increase of secondary terminal voltage when the load is disconnected (primary impressed volt- age and frequency remaining unaltered). The connection between magnetic leakage and voltage regulation will be studied by considering the simplest possible cases, and noting the difference in secondary flux-linkages under loaded and open-circuited conditions. The amount of the leakage flux in proportion to the useful flux will purposely be greatly exaggerated, and, in order to eliminate unessential considerations, the fol- lowing assumptions will be made : (1) The magnetizing component of the primary cur- rent will be considered negligible relatively to the total current, and will not be shown in the diagrams. (2) The voltage drop due to ohmic resistance of both primary and secondary windings will be neglected. (3) The primary and secondary windings will be sym- metrically placed and will consist of the same number of turns. (4) One flux line as shown in the diagrams linking with one turn of winding will generate one volt. In Fig. 35, both primary and secondary coils consist of one turn of wire wound close around the core: a cur- rent 7 S is drawn from the secondary on a load of power factor cos 0, causing a current /i exactly equal but opposite to Is to flow in the primary coil, the result being the leakage flux as represented by the four dotted lines. 110 PRINCIPLES OF TRANSFORMER DESIGN The secondary voltage, E s = 2 volts, is due to the two flux lines which link both with the primary and secondary condary FIG. 35. Magnetic Leakage: Thickness of Coils Considered Negligible. coils. The phase of this component of the total flux is, therefore, 90 in advance of E s as indicated by the line OB in the vector diagram. MAGNETIC LEAKAGE IN TRANSFORMERS 111 In order to calculate the necessary primary impressed e.m.f., we have as one component OE'\ exactly equal but opposite to OE S because the flux OB will induce in the primary coil a voltage exactly equal to E s in the second- ary. The other component is E f \E v = ^. volts, equal but opposite to the counter e.m.f. which, being due to the four leakage lines created by the current /i, will lag 90 in phase behind OI\. The resultant is OE P which scales 5 volts.* When load is thrown off the transformer there will be five lines linking with the primary which, since there is now no secondary m.m.f. to produce leakage flux, will pass through the iron core and link with the secondary. The secondary voltage on open circuit will, therefore, be E p = 5 volts, and the percentage regulation is EpE s 5 2 looX ^ = iooX = 150. <<! 2 In Fig. 36, a departure is made from the extreme simplicity of the preceding case in order to illustrate the effect of leakage lines passing not only entirely outside the windings, but also through the thickness of the coils, as must always happen in practical transformers where the coils occupy an appreciable amount of space. Each winding now consists of two turns, with an air space between the turns through which leakage flux * The reason why the six flux lines shown in the figure as linking with the primary coil do not generate 6 volts is, of course, due to the fact that these flux lines are not all in the same phase; the resultant or actual flux in the core under the primary coil is 5 lines, as indicated by the vector diagram. The actual amount of flux passing any given cross-section of the core must be thought of as the (vectorial) addition of the flux lines shown in the sketch at that particular section. 112 PRINCIPLES OF TRANSFORMER DESIGN represented in Fig. 36 by one dotted line is supposed to pass. The single flux line, linking with both the primary s~ X Secor V B dary Hi i -, ::rr- v > I t t s> i @ i i s \ / l v ~: . _' V FIG. 36. Magnetic Leakage: Thickness of Coils Appreciable. and secondary windings, generates the e.m.f. component E = 2 volts. The leakage flux line marked F links with MAGNETIC LEAKAGE IN TRANSFORMERS 113 only one turn of the secondary, and therefore generates one volt lagging 90 in phase behind the primary current /i. The total secondary voltage is E s which scales 2.6 volts; the balancing component in the primary being EI. It should be particularly noted that this balancing component does not account for the full effect of the two flux lines B and F linking with the primary, because, while the flux line F links with only one secondary turn, it links with two primary turns. The voltage com- ponent OE'i in the primary may, therefore, be thought of as due to the flux lines B and /, leaving for the remain- ing component of the impressed e.m.f., E\E P = 6 volts (leading O/i by 90) which may be considered as caused by the three lines F, H, and G. In other words, the reactive drop (I\X P ) depends upon the di/erence between the primary and secondary flux-linkages of the stray magnetic field set up by the combined action of the secondary current I s and the balancing component /i of the total primary current. (In this case /i is the total primary current, since the magnetizing component is neglected.) The leakage flux-linkages are as follows: With the primary turns : JXi = i volt, HX2 = 2 volts, GX2 = 2 volts, FX2 = 2 volts 7 volts With the secondary turns: FXi = i volt giving a difference of 6 volts 114 PRINCIPLES OF TRANSFORMED DESIGN This is the. vector (I\X P }. When applying this rule T to actual transformers in which the ratio of turns ~ is 1 s not unity, the proper correction must be made (as ex- plained later) when calculating the equivalent e.m.f. component in the primary circuit. To obtain the regulation in the case of Fig. 36, we have E P = S volts and E s = 2.6 volts when the transformer is loaded. When the load is thrown off, there will be four flux lines linking with both primary and secondary producing 8 volts in each winding. The regulation is therefore, 8-2.6 100 X ^- = 208 per cent. 2.0 33. Experimental Determination of the Leakage Reac- tance of a Transformer. Although these articles are written from the viewpoint of the designer, who must predetermine the performance of the apparatus he is designing, a useful purpose will be served by considering how the leakage reactance of an actual transformer may be determined on test. The purpose referred to is the clearing up of any vagueness and consequent inaccuracy that may exist in the mind of the reader, due largely in the writer's opinion to the common, but unnecessary if not misleading assumption, that the secondary has self induction.* * The assumption usually made in text books is that the secondary self-induction (i.e., the flux produced by the secondary current, and linking with the secondary turns) is equal to the primary leakage self-induction, MAGNETIC LEAKAGE IN TRANSFORMERS 115 The diagram, Fig. 37, shows the secondary of a transformer short-circuited through an ammeter, A, of negligible resistance. The impressed primary voltage E z , of the frequency for which the transformer is de- signed, is adjusted until the secondary current I s is indicated by the ammeter. If the number of turns in the primary and secondary are T p and T s respectively, IT \ the primary current will be 7i=/ s ( M because, the \7 p/ amount of flux in the core being very small, the mag- FIG. 37. Diagram of Short-circuited Transformer. netizing component of the primary current may be neglected. The measured resistances RI and R 2 of the primary and secondary coils being known, the vector diagram Fig. 38, can be constructed. The volts induced in the secondary are OE% (equal to I S R2) in phase with the current I s . The bal- ancing component in the .primary winding is OE'\ equal to 2 ( ~ j in phase with the primary current I\. Another component in phase with this current is E'\P (equal to I\R\). Since the total impressed voltage 116 PRINCIPLES OF TRANSFORMER DESIGN has the known value E z , we can describe an arc of circle of radius OE Z from the point as a center. By erecting a perpendicular to O/i at the point P, the point E z is determined, and E Z P is the loss of pressure caused by magnetic leakage. The vector OP may be thought of as the product of the primary current /i, FIG. 38. Vector Diagram of Short-circuited Transformer. and an equivalent primary resistance R p , which assumes the secondary resistance to be zero, but the primary resistance to be increased by an amount equivalent to the actual secondary resistance. Thus, but TA J.r MAGNETIC LEAKAGE IN TRANSFORMERS 117 whence, ...... (35) In order to get an expression for the transformer leakage reactance (X p ) in terms of the test data, we can write, whence This quantity, multiplied by /i (or IiX p = VE z 2 -(IiR P ) 2 ) is the vector E'iE p of the diagrams in Figs. 35 and 36 as it might be determined experi- mentally for an actual transformer. If it were possible for all the magnetic flux to link with all the primary, and all the secondary, turns, the quantity IiX P would necessarily be zero; all the flux would be in the phase OB, and OE Z (of Fig. 38) would be equal to OP. The presence of the quantity IiX P can only be due to those flux lines which link with primary turns, but do not link with an equivalent number of secondary turns. 34. Calculation of Reactive Voltage Drop. Seeing that it is generally although not always desirable to obtain good regulation in transformers, it is obvious that designs with the primary and secondary windings on separate cores (see Figs, i, 35 and 36), which greatly exaggerate the ratio of leakage flux to useful flux, would be very unsatisfactory in practice. By putting half the primary and half the secondary on each of the two limbs of a 118 PRINCIPLES QF TRANSFORMER DESIGN single-phase core-type transformer, as shown in Fig. 39, a considerable improvement is effected, but the reluctance of the leakage paths is still low, and this design is not nearly so good as Fig. 7 (page 18) where the leakage paths have a greater length in proportion to the cross-section. Similarly in the shell type of transformer, the design shown in Fig. 40 is unsatisfactory; the arrangement of FIG. 39. Leakage Flux Lines in Special Core-type Transformer. coils, as shown in Figs. 10 and n (Art. 8) is much better because of the greater reluctance of the leakage paths. Transformers with coils arranged as in Figs. 7 and 10 are satisfactory for small sizes; but, in large units, it is neces- sary to subdivide the windings into a large number of sections with primary coils " sandwiched " between secondary coils as in Fig. 17 (core type) and Figs. 8. and MAGNETIC LEAKAGE IN TRANSFORMERS 119 1 6 (shell type). By subdividing the windings in this manner, the m.m.f. producing the leakage flux, and the number of turns which this flux links with, are both greatly reduced. The objection to a very large number of sections is the extra space taken up by insulation between the primary and secondary coils. For the purpose of facilitating calculations, the windings of transformers can generally be divided into unit sections FIG. 40. Leakage Flux Lines in Poorly Designed Shell-type Transformer. as indicated in Fig. 41 (which shows an arrangement of coils in a shell- type transformer similar to Fig. 16). Each section consists of half a primary coil and half a secondary coil, with leakage flux passing through the coils and the insulation between them * all in the same * If air ducts are required between sections of the winding, these should be provided in the position of the dotted center lines, by a further sub- division of each primary and secondary group of turns; thus allowing the space between primary and secondary coils to be filled with solid insulation. It is evident that, if good regulation is desired, the space between primary and secondary coils where the leakage flux density has its maximum value must be kept as small as possible. 120 PRINCIPLES OF TRANSFORMER DESIGN direction, as indicated by the flux diagram at the bottom of the figure. Unit section.. I FIG. 41. Section through Coils of Shell-type Transformer. The effect of all leakage lines in the gap between the coils is to produce a back e.m.f. in the primary without MAGNETIC LEAKAGE IN TRANSFORMERS 121 affecting the voltage induced in the secondary by the main component of the total flux (represented by the full line). Of the other leakage lines, B links with only a portion of the primary turns and has no effect on the primary turns which it does not link with ; while A links not only with all the primary turns, but also with a cer- tain number of secondary turns. Note that if the line A were to coincide with the dotted center line MN, marking the limit of the unit section under consideration, it would have no effect on the transformer regulation because flux which links equally with primary and secondary is not leakage flux. Actually, the line A links with all the primary turns of the half coil in the section considered, but with only a portion of the second- ary turns in the same section, Its effect is, therefore, exactly as if it linked with only a fractional number of the primary turns. The mathematical .development which follows is based on these considerations. Fig. 42 is an enlarged view of the unit section of Fig. 41, the length of which measured perpendicularly to the cross section -is / cms.- All the leakage is supposed to be along parallel lines perpendicular to the surface of the iron core above and below the coils. It is desired to calculate the reactive voltage drop in a section of the winding of length / cms., depth h cms., and total width (s+g+p) cms., where s = the half thickness of the secondary coil; g = the thickness of insulation between primary and secondary coils; p = the half thickness of the primary coil. 122 PRINCIPLES OF TRANSFORMER DESIGN The voltage drop caused by the leakage flux in the spaces g, p, and s will be calculated separately and then added together to obtain the total reactive voltage drop. The general formula giving the r.m.s. value of the volts induced by < maxwells linking with T turns is (36) when the flux variation follows the simple harmonic law. FIG. 42. Enlarged Section through Transformer Coils. In calculating the voltage produced by a portion of flux in a given path, we must therefore determine (i) MAGNETIC LEAKAGE IN TRANSFORMERS 123 the amount of this flux, and (2) the number of turns with which it links. The symbols T\ and TI will be used to denote the number of turns in the half sections of widths p and 5 of the primary and secondary coils respectively. The meaning of the variables x and y is indicated in Fig. 42. The symbol m will be used for 27T/" the quantity -^. For the section g we have, Inserting for 3> its value in terms of m.m.f. and per- meance, this becomes," (7^ = ^(04^70 X^XTY . . (37) In the section p, the m.m.f. producing the element of flux in the space of width dx is due to the current 7i in (- }Ti turns, and since this element of flux links . /*\ in ( - w (x\ - ) T\ turns, we have, whence f. (38) 124 PRINCIPLES OF TRANSFORMER DESIGN In the section s, the m.m.f., producing the small element of flux in the space of width dy is due to the current 7, in (-}TI turns, and since this must be considered as linking with ( i ) T\ turns, we can write, whence s 2 h */ \j i- (39) wherefrom the -secondary quantities T 2 and I s have been eliminated by putting (Ti/i) in place of (TWs). The final expression for the inductive voltage drop in the unit section considered is obtained by adding together the quantities (37), (38), and (39). Thus, ' (40) wherein all dimensions are expressed in centimeters. If all the primary turns are connected in series, this T quantity will have to be multipled by the ratio - (or Ii by twice the number of primary groups of coils) to obtain the value of the vector I\ X p shown in the vector diagrams. MAGNETIC LEAKAGE IN TRANSFORMERS 125 Equivalent Value of the Length I. The numerical value of the length / as used in the above formulas might reasonably be taken as the mean length per turn of the transformer windings, provided the reluctance of the flux paths outside the section shown in Fig. 42 may be neglected, not only where the iron laminations provide an easy path for the flux, but also where the ends of the coils project beyond the stampings. Every manufacturer of transformers who has accu- mulated sufficient test data from transformers built to his particular designs, will be in a position to modify Formula (40) in order that it may accord very closely with the measured reactive voltage drop. This correc- tion may be in the form of an expression for the equiva- lent length /, which takes into account the type of transformer (whether core or shell) and the arrange- ment of coils; or the quantity g + may be modified, being perhaps more nearly g+ , which allows for more leakage flux through the space occupied by the copper than is accounted for on the assumption of parallel flux lines. The writer believes, however, that if / is taken equal to the mean length per turn of the windings expressed in centimeters the Formula (40) will yield results sufficiently accurate for nearly all practical purposes. 35. Calculation of Exciting Current. Before drawing the complete transformer vector diagram, including the reactive drop calculated by means of the formula devel- oped in the preceding article, it is necessary to consider 126 PRINCIPLES OF TRANSFORMER DESIGN how the magnitude and phase of the exciting current component of the total primary current may be pre- determined. The exciting current (TV) may be thought of as con- sisting of two components: (i) the magnetizing com- ponent (/o) in phase with the main component of the magnetic flux, i.e., that which links with both primary and secondary coils, and (2) the " energy " component Max 4 value of current component Amp, turns to produoe B max. Phaee of induced e. m.f. Total iron loss (watts) Primary impressed volts. FIG. 43. Vector Diagram showing Components of Exciting Current. (I w ) leading /o by one-quarter period, and, therefore, exactly opposite in phase to the induced e.m.f. The magnitude of this component depends upon the amount of the iron losses only, because the very small copper losses (I 2 e Ri) may be neglected. If these components could be considered sine waves, the vector construction of Fig. 43 would give correctly the magnitude and phase of the total exciting current L. For values of flux density above the " knee " of the B- H curve, the instantaneous values of the magnetizing current are no longer proportional to the flux, and this MAGNETIC LEAKAGE IN TRANSFORMERS 127 component of the total exciting current cannot therefore be regarded as a sine wave even if the flux variations are sinusoidal. The error introduced by using the con- struction of Fig. 43 is, however, usually negligible because the exciting current is a very small fraction of the total primary current. The notes on Fig. 43 are self explanatory, but reference should be made to Fig. 44 from which the ampere turns per inch of the iron core may be read for any value of the (maximum) flux density. The flux density is given in gausses, or maxwells per square centimeter of cross- section.* The total magnetizing ampere turns are equal to the number read off the curve multiplied by the mean length of path of the flux which links with both primary and secondary coils. When butt joints are present in the core, the added reluctance should be allowed for. Each butt joint may be considered as an air gap 0.005 m - long, and the ampere turns to be allowed in addition to those for the iron portion of the magnetic circuit are therefore, HI A mp. turns for joints O.47T = 9.01 X m ax XNo. of butt joints in series. (41) Instead of calculating the exciting current by the method outlined above, designers sometimes make use * The writer makes no apology for using both the inch and the centi- meter as units of length. So long as engineers insist that the inch has certain inherent virtues which the centimeter does not possess, they should submit without protest to the inconvenience ancl possible dis- advantage of having to use conversion factors, especially in connection with work based on the fundamental laws of physics. 128 PRINCIPLES OF TRANSFORMER DESIGN 16000 14000 13000 12000 &11000 10000 9000 8000 7000 6000 5000 10 20 30 40 50 Ampere-turns per inch GO 70 FIG. 44. Curve giving Connection between Magnetizing Ampere-turns and Flux Density in Transformer Iron. MAGNETIC LEAKAGE IN TRANSFORMERS 129 of curves connecting maximum core density and volt- amperes of total exciting current per cubic inch or per pound of core; the data being obtained from tests on completed transformers. The fact that the total volt- amperes of excitation (neglecting air gaps) are some function of the flux density multiplied by the weight of the iron in the transformer core, may be explained as follows : Let w = total watts lost per pound of iron, correspond- ing to a particular value of B as read off one of the curves of Fig. 27; a = Ampere turns per inch as read off Fig. 44 ; A = cross-section of iron in the core, measured perpendicularly to the magnetic flux lines (square inches) ; /= Length of the core in the direction of the flux lines (inches) ; P = Weight of core in pounds = o.2&Al. The symbols previously used are: T p = number of primary turns: Given definite values for B and /, the " in phase " component of the exciting current is T core loss wXP 130 PRINCIPLES OF TRANSFORMER DESIGN and the " wattless " component, or true magnetizing current, is aXl' whence Ie Multiplying both sides of the equation by ~, we get Epl e _ volt-amperes of total excitation P weight of core < This formula may be used for plotting curves such as those in Fig. 45. Thus, if = 13,000 gausses, /=6o cycles per second, 20 = 1.55 (read off curve for silicon steel in Fig. 27), a = 22 (from Fig. 44); and, by Formula (42) Volt-amperes per pound The error in this method of deriving the curves of Fig. 45 is due to the fact that sine waves are assumed. The data for plotting the curves should properly be obtained from tests on cores made out of the material to be used in the construction of the transformer. MAGNETIC LEAKAGE IN TRANSFORMERS 131 16000 15000 14000 13000 12000 * 11000 3 10000 I I 9000 8000 7000 GOOO 5000 5 10 15 20 25 30 35 40 Exciting volt-aiperes per Ib. of stampings (Approximate values for either iron or silicon-steel) FIG. 45. Curves giving Connection between Exciting Volt-amperes and Flux Density in Transformer Stampings. 132 PRINCIPLES OF TRANSFORMER DESIGN The effect of the magnetizing current component in distorting the current waves may be appreciable when the core density is carried up to high values. The curve of flux variation cannot then be a sine wave, and the introduction of high harmonics in the current wave may aggravate the disturbances that are always liable to occur in telephone circuits paralleling overhead trans- mission lines. This is one reason why high values of the exciting current are objectionable. An open-circuit primary current exceeding 10 per cent of the full-load current would rarely be permissible. 36. Vector Diagrams Showing Effect of Magnetic Leakage on Voltage Regulation of Transformers. The t vector diagrams, Figs. 46, 47, and 48, have been drawn to show the voltage relations in transformers having appreciable magnetic leakage. The proportionate length of the vectors representing IR drop. IX drop, and magnetizing current, has purposely been exaggerated in order that the construction of the diagrams may be easily followed. Fig. 46 is the complete vector diagram of a transformer ; the meaning of the various component quantities being as follows: 2 = Induced secondary e.m.f., due to the flux (OB) linking with the secondary turns; E s = Secondary terminal voltage when the secondary current is I s amperes on a load power factor of cos 6] I e = Primary exciting current, calculated as ex- plained in the preceding article; MAGNETIC LEAKAGE IN TRANSFORMERS 133 /i = Balancing component of total primary current IP = Total primary current ; E'\ = Balancing component of induced primary volt- PE'\ = IR drop due to primary resistance (drawn parallel to 01 P ) ; EpP = IX drop due to leakage reactance (drawn at right angles to OI P ) ; E v Impressed primary e.m.f . FIG. 46. Vector Diagram of Transformer on Inductive Load. It is usually permissible to neglect the exciting current component when considering full-load conditions. This leads to the simpler diagram, Fig. 47, in which the total primary current is supposed to be of the same magnitude and phase as what has previously been referred to as the balancing component of the total primary current. The dotted lines in Fig. 47 show how a still greater simplification may be effected in drawing a vector 134 PRINCIPLES OF TRANSFORMER DESIGN diagram from which the voltage regulation can be cal- culated. Instead of drawing the two vectors QEi and OE S for the induced and terminal secondary voltages, we can draw OE ( opposite in phase to and equal to Then E e P (drawn parallel to 01 1) is the component of the impressed primary volts necessary to overcome the ohmic resistance of both primary and secondary windings. Equivalent Secondary Risistance >- dropj FIG. 47. Simplified Vector Diagram of Transformer; Exciting Current Neglected. It is now only necessary to turn this diagram through 1 80 degrees, and eliminate all unnecessary vectors, in order to arrive at the very simple diagram of Fig. 48, from which the voltage regulation can be calculated. 37. Formulas for Voltage Regulation. From an inspection of Fig. 48, it is seen that (IiR p )+EeCasO COS <b . . (43) MAGNETIC LEAKAGE IN TRANSFORMERS 135 wherein cos is known (being the power factor of the external load), and cos <f> has not yet been determined. But, sinfl ( . * ' ' (44) FIG. 48. Simple Transformer Vector Diagram for Calculation of Voltage Regulation. which can be used to calculate and therefore cos>. The percentage regulation is , (45) or, if the ohmic drop is expressed as a percentage of the (lower) terminal voltage: Per cent regulation _Per cent equiv. IR drop +100 (cos 9 cos #) / ,\ COS0 136 PRINCIPLES OF TRANSFORMER DESIGN The difference between the angles 6 and (Fig. 48) is generally small, and it is then permissible to assume that OD = OE P . But E e +IiR cos d+IiX p sin 6, whence, Per cent regulation (approximate) = Per cent IR cos 0+per cent IX sin 0. (47) If the power factor were leading instead of lagging as in Fig. 48, the plus sign would have to be changed to a minus sign. Example. In order to show that the approximate Formula (47) is sufficiently accurate for practical pur- poses, the following numerical values are assumed. Power factor (cos 0) =0.8. Total IR drop = 1.5 per cent. Total IX drop = 6.0 per cent. By Formula (44), 0.06+0.6 tan<= =0.81, 0.015+0.8 whence cos <f> = 0.777, an d> by Formula (46), Regulation = '-U- = 4.9 per cent. 0.777 By the approximate Formula (47), Regulation = (i.sXo.8) + (6Xo.6)=4.8 per cent. MAGNETIC LEAKAGE IN TRANSFORMERS 137 The total equivalent voltage drop, due to the resistance of the windings (the quantity I\R P of the vector dia- grams) is usually between i and 2 per cent of the ter- minal voltage in modern transformers. The reactive voltage drop caused by magnetic leakage (the quantit}^ IiXp in the vector diagrams) is nearly always greater than the IR drop, being 3 to 8 per cent of the terminal voltage. Sometimes it is TO per cent, or even more, especially in high- voltage transformers where the space occupied by insulation is considerable, or in transformers of very large size, when the object is to keep the current on short circuit within safe limits. CHAPTER V PROCEDURE IN TRANSFORMER DESIGN 37. The Output Equation. The volt-ampere output of a single-phase transformer is E X / which, as explained in Art. 6, may be written Volt-amperes = x$X(TI), . . (48) where TI stands for the total ampere turns of either the primary or secondary winding. There is no limit to the number of designs which will satisfy this equation; the total flux, 3>, is roughly a measure of the cross-section of the iron core, while the quantity (TI) determines the cross-section of the wind- ings. The problem before the designer is to proportion the parts and dispose the material in such a way as to obtain the desired output and specified efficiency at the lowest cost. The temperature rise is also a matter of importance which must be watched, and light weight is occasionally more important than cost. It cannot be said that there is one method of attacking the problems of transformer design which has indisputable advantages over all others; and in this, as in all design, the judgment and experience of the individual designer must necessarily play an important part. The apparent 138 PROCEDURE IN TRANSFORMER DESIGN 139 simplicity of the calculations involved in transformer design is the probable cause of the many more or less unsuccessful attempts to reach the desired end by purely mathematical methods. It is not possible to include all the variable factors in practical mathematical equa- tions purporting to give the ideal quantities and pro- portions to satisfy the specification. Methods of pro- cedure aiming to dispense with individual judgment and a certain amount of correction or adjustment in the final design, should generally be discountenanced, because they are based on inade'quate or incorrect assumptions which are liable to be overlooked as the work proceeds and becomes finally crystallized into more or less for- midable equations and formulas of unwieldy propor- tions. No claim to originality is made in connection with the following method of procedure; indeed it is ques- tionable whether the mass of existing literature treating of the alternating current transformer leaves anything new to be said on the subject of procedure in design. All that the present writer hopes to present is a treat- ment consistent with what has gone before, based always on the fundamental principles of physics even though the use of empirical constants may be necessary. Instead of attempting to take account at one time of all the conditions to be satisfied in the final design, the factors which have the greatest influence on the dimen- sions will be considered first; items such as temperature rise and voltage regulation being checked later and, if necessary, corrected by slight changes in the dimensions or proportions of the preliminary design. 140 PRINCIPLES OF TRANSFORMER DESIGN 38. Specifications. It will be advisable to list here the particulars usually specified by the buyer, and sup-, plement these, if required, with certain assumptions that the manufacturer must make before he can proceed with a particular design. (1) K.v.a. output. (2) Number of phases. (3) Primary and secondary voltages (E p and E s ). (4) Frequency (/). (5) Efficiency under specified* conditions. (6) Voltage regulation under specified load. (7) Method of cooling Temperature rise. (8) Maximum permissible open-circuit exciting cur- rent. Items (i) to (4) must always be stated by the pur- chaser, while the other items may be determined by the manufacturer, who should, however, be called upon to furnish these particulars in connection with any competi- tive offer. With reference to item (5), if the efficiency is stated for two different loads, the permissible copper and iron losses can be calculated. If the buyer does not furnish these particulars, he should state whether the trans- former is for use in power stations or on distributing lines, in order that the relation of the iron losses to the total losses may be adjusted to give a reasonable all-day efficiency. In any case, before proceeding with the design, the maximum permissible iron and copper losses must be known or assumed. The requirements of items (6), (7). and (8), are to some PROCEDURE IN TRANSFORMER DESIGN 141 extent satisfied, even in the preliminary design, by selecting a flux density (B) and a current density (A) from the values given in Article 20, because industrial competition and experience have shown these values to give the best results while using the smallest per- missible amount of material. Thus, by selecting a proper value for A, both the local heating and the IR drop of the windings will probably be within reason- able limits. The other factor influencing the voltage regulation (item (6)) is the reactive drop, which can generally be controlled by suitably subdividing the windings. A proper value of the flux density (B) will generally keep item (8) within the customary limits. 39. Estimate of Number. of Turns in Windings. Re- turning to the Formula (48) in Article 37, if a suitable value for T could be determined or assumed, the only unknown quantity in the output equation would be 3> and we should then have a starting-point from which the dimensions of a preliminary design could be easily cal- culated. Let T^_= volts per turn (of either primary or sec- ondary winding) then, in order to express this quan- tity in terms of the volt-ampere output, we have, TI' from which T must be eliminated, since the reason for seeking a value for V t is that T may be calculated therefrom. 142 PRINCIPLES OF TRANSFORMER DESIGN Using the value of (El) as given by Formula (48), we can write TI TYXio 8 ' whence V t = Vvolt-ampere output X - (49) The quantity in brackets under the second radical is found to have an approximately constant value, for an efficient and economical design of a given type, without reference to the output. This permits of the formula being put in the form V t = cX Vvolt-ampere output, .... (490) where c is an empirical coefficient based on data taken from practical designs. Factors Influencing the Value of the Coefficient c. \ f ^ It is proposed to examine the meaning of the ratio which appears under the second radical of Formula (49) with a view to expressing this in terms of known quan- tities, or of quantities that can easily be estimated. Let W c = full load copper losses (watts) ; Wi = core losses (watts) ; the relation between these losses being; (So) PROCEDURE IN TRANSFORMER DESIGN 143 wherein b must always be known before proceeding with the design. Let l c = mean length per turn of copper in windings ; /i = mean length of magnetic circuit measured along flux lines; then W c = constant X A 2 X volume of copper i where k c is a constant to be determined later. Similarly Wi = constant XfB n Xvolume of iron (52) wherein k L is another constant to be determined later. Inserting these values in Formula (50), the required ratio can be put in the form ( } TI bk,B n - l \l This ratio is thus seen to depend on certain quantities and constants which are only slightly influenced by the output of the transformer. They -depend on such items as the ratio of copper losses to iron losses (i.e., whether the transformer is for use on power transmission lines, or distributing circuits); temperature rise and methods of 144 PRINCIPLES OF TRANSFORMER DESIGN cooling; space factor (voltage); and also on the type whether core or shell since this affects the best relation between mean lengths of the copper and iron circuits. The Factor k c . Using the inch for the unit of length, and allowing 7 per cent for eddy-current losses in the copper, the resistivity of the windings will be 0.9X10" ohms per inch-cube at a temperature of 80 C.; the loss per cubic inch of copper = A 2 X 0.9X10-, and ITI\ since the volume is 2( j/ c , it follows that k c = 2X o.9Xio~ 6 . The Factor ki If w = total watts lost per pound as read off one of the curves of Fig. 27, and if h is in inches, we have the equation whence 0.28^ 6-45/5" The Factor b. The ratio of full-load copper loss to iron loss will determine the load at which maximum efficiency occurs. Let us assume the k.v.a. output and the frequency of a given transformer to be constant, and determine the conditions under which the total losses will be a minimum. It is understood that, if the current / is increased, the voltage. E, must be decreased; but the condition k.v.a, = El must always be satisfied. PROCEDURE IN TRANSFORMER DESIGN 145 The sum of the losses is PFc+H^; but (k.v.a.) 2 a constant 2 2 and Also, since/ remains constant, EccB, and we can write Wi = a constant X# n . The quantity which must be a minimum is therefore a constant . . ha constantX-E . If we take the differential coefficient of this function of E and put it equal to zero, we get the relation W~ 2 The value of n for high densities is about 2, while for low densities it is nearer to 1.7, a good average being 1.85. Thus, to obtain maximum efficiency at full load in a power transformer, the ratio of copper loss to iron loss should be about b = -^-- =0.925. In a distributing transformer, in order to obtain a good all-day efficiency, the maximum efficiency should occur at about f full load, whence W t 146 PRINCIPLES OF TRANSFORMER DESIGN Taking 71 = 1.75, because of the lower densities generally used in small self-cooling transformers, we get , 1.75X9 / x b= -=61.97 or (say) 2. 4X2 The Ratio -. Considerable variations in this ratio are It permissible, even in transformers of a given type wound for a particular voltage, and that is one reason why a close estimate of the volts per turn as given by Formula (49) is* not necessary. Refinements in proportioning the dimensions of a transformer are rarely justified by any appreciable improvement in cost or efficiency; a certain minimum quantity of material is required in order to keep the losses within the specified limits; but consid- erable changes in the shape of the magnetic and electric circuits can be made without greatly altering the total cost of iron and copper, provided always that the im- portant items of temperature rise and regulation are checked and maintained within the specified limits. Figs. 49 and 50 show the assembled iron stampings of single-phase shell- and core- type transformers. The proportions will depend somewhat upon the voltage and method of cooling; but if the leading dimensions are expressed in terms of the width (L) of the stampings under the coils, they will generally be within the following limits: Shell T3>pe. Core Type. S 2 to 3 times L S = i to i . 8 times L B = o.$ to o . 75 times L B = i to i . 5 times L D = o.6 to 1.2 times L D = i to 2 times L H = i . 2 to 3 5 times L H = 3 to 6 times L PROCEDURE IN TRANSFORMER DESIGN 147 By taking the averages of these figures, and roughly approximating the lengths l c and k in each case, the mean value cf the required ratio is found to be " 7 = 1.2 (approx.) for shell type, ] k 7 = 0.3 (approx.) for core type. FIG. 4Q. Assembled Stampings of Single-phase Shell-type Transformer. Having determined the values of the various quan- tities appearing in Formula (53), it is now possible to calculate an approximate average value for the fa quantity ^ and for the coefficient c of Formula (49). We shall make the further assumptions (refer Art. 20) that A = 1100 amperes per square inch, and ^=8000 148 PRINCIPLES OF TRANSFORMER DESIGN gausses; the transformer being of the shell type for use on distributing circuits of frequency 60. Then, by Formula (53), ^ g /$ _^<o79X i IPO X i. 2X6.45X60X9000 _ ~Ti~ ~io 6 x 2x0.28x0.75 = 19,720 FIG. 50. Assembled Stampings of- Single-phase Core-type Transformer. wherein the figure 0.75 is the value of w read off the curve for silicon steel in Fig. 27. The value of the coefficient in Formula (49), for the assumed conditions, is therefore -^ X 19, 720 = 0.0296 PROCEDURE IN TRANSFORMER DESIGN 149 Similarly, for a core-type power transformer; if 7=25. B = 13,000, and A = 1350, we have, /$ = 2X0.9X1350X0.5X6.45X25X13,000^ 6 TI io 6 Xo.925Xo.28Xo.58 Whence c = 0.02 74. Having shown what factors determine this design coefficient, it will merely be necessary to give a list of values from which a selection should be made for the purpose of calculating the quantity V t of Formula (490).- For shell-type power transformers = 0.04 to 0.045 For shell- type distributing transformers ^ = 0.03 For core- type power transformers = 0.025 to 0.03 For core- type distributing transformers c = o.o2 Where a choice of two values of c is given, the lower value should be chosen for transformers wound for high pressures. When the voltage is low the value of c is slightly higher because of the alteration in the ratio. - which depends somewhat on the copper space factor. li The proposed values here given for this design coeffi- cient are based on the assumption that silicon steel stampings are used in the core. If ordinary trans- former iron is used as, for instance, in small distrib- uting transformers it will be advisable to take about J of the above values for the coefficient c. 40. Procedure to Determine Dimensions of a New Design. With the aid of the design coefficient c, it is now possible to calculate the number of volts that should 150 PRINCIPLES OF TRANSFORMER DESIGN be generated in one turn jf the winding of a transformer of good design according to present knowledge and prac- tice. The logical sequence of the succeeding steps in the design, may be outlined as follows: (1) Determine approximate dimensions. (a) Calculate volts per turn by Formula (49). (b) Assume current density (select suitable trial value from table in Art. 20). Decide on number of coils. Calculate cross-section of copper. (c) Decide upon necessary insulation and oil- or air-ducts between coils, and between windings and core. Determine shape and size of " window " or opening necessary to accommodate the windings. (d) Calculate total flux required. Assume flux density (select suitable trial value from table in Art. 20), and calculate cross-sec- tion of core. Decide upon shape and size of section, including oil- or air-ducts if necessary. (e) Calculate iron and copper losses, and modify the design slightly if necessary to keep these within the specified limits. (2) Calculate approximate weight and cost of iron and copper if desired to check with permissible maximum before proceeding with the design. (3) Calculate exciting current. (4) Calculate leakage reactance and voltage regula- tion. PROCEDURE IN TRANSFORMER DESIGN 151 . (5) Calculate necessary cooling surfaces. Design con- taining tank and lid, providing not only sufficient oil capacity and cooling surface, but also the necessary clearances to insure proper insulation between current- carrying parts and the case. Calculate temperature rise. 41. Space Factors. The copper space factor, as pre- viously defined (see Art. 15), is the ratio between the cross-section of copper and the area of the opening or " window " which is necessary to accommodate this copper together with the insulation and oil- or air-ducts. It may vary" between 0.55 in transformers for use on circuits not exceeding 660 volts, to 0.06 in power trans- formers wound for about 100,000 volts. An estimated value of the probable copper space factor may be useful to the designer when deciding upon one of the dimensions of the " window " in the iron core. For this purpose, the curves of Fig. 51 may be used, although the best design and arrangement of coils and ducts will not always lead to a space factor falling within the limits included between these two curves. Iron Space Factor. The so-called stacking factor for the iron core will be between 0.86 and 0.9, and the total thickness of core, multiplied by this factor, will give the net thickness of iron if there are no oil- or air-ducts. When spaces are left between sections of the core for air or oil circulation, the iron space factor may be from 0.65 to 0.75. 42. Weight and Cost of Transformers. The weight per k.v.a. of transformer output depends not only upon the total output, but also upon the voltage and fre- quency. The net and gross weights of particular trans- 152 PRINCIPLES OF TRANSFORMER DESIGN S| \ I : PROCEDURE IN TRANSFORMER DESIGN 153 formers can be obtained from manufacturers' catalogues and also from the Handbooks for Electrical Engineers. The effect of output and frequency on the weight of a line of transformers designed for a particular voltage (in this instance, 22,000 volts) is roughly indicated by the following figures of weight per k.v.a. of output. These figures include the weight of oil and case. { 100 k.v.a. output 40 Ib. Frequency 60 \ [ 500 k.v.a. output 23 Ib. ^ f 100 k.v.a. output 152 Ib. Frequency 25 \ [ 500 k.v.a. output 35 Ib. The cost of transformers, depending as it does on the fluctuating prices of copper and iron, is very unstable. Within the last few years, the variation in the price of copper wire has been about 100 per cent, and the cost of the laminated iron for the cores has also undergone great changes. The best that can be done here is to indicate how the cost depends upon voltage and output. That a high frequency always means a cheaper transformer is evident from an inspection of the fundamental Formula (48) of Art. 37. If/ is increased, either 3>, or (TI), or both, can be reduced, and this means a saving of iron, or copper, or both. The effect of an increase in voltage is felt particularly in the smaller sizes, but an increase of voltage always means an addition to the cost; while an increase of size for a given voltage results in a reduc- tion of the cost per k.v.a. of output. Some idea of the dependence of cost on output and voltage may be gained from the fact that the unit cost 154 PRINCIPLES OF TRANSFORMER DESIGN would be about the same for (i) a 1500 k.v.a. trans- former wound for 22,000 volts, (2) a 2000 k.v.a. trans- former wound for 44,000 volts, and (3) a 3000 k.v.a. transformer wound for 88,000 volts. Three-phase Transformers. It does not appear to be necessary to supplement what has been said in Articles 5 and 8 on the subject of three-phase transformers. Once the principles underlying the design of single-phase transformers are thoroughly understood, it is merely necessary to divide any polyphase transformer (see Figs. 12, 13, and 14) into sections which can be treated as single-phase transformers, due attention being paid to the voltage and k.v.a. capacity of each such unit section of the three-phase transformer. The saving of materials effected by combining the magnetic circuits of three single-phase transformers so as to produce one three-phase unit, usually results in a reduction of 10 per cent in the weight and cost. 43. Numerical Example. It is proposed to design a single-phase 1500 k.v.a. oil-insulated, water-cooled, transformer for use on an 88,000- volt power transmission system. A design sheet containing more detailed items than would generally be considered necessary will be used in order to illustrate the various steps in the design as developed and discussed in the preceding articles. Two columns will be provided for recording the known or calculated quantities, the first being used for preliminary assumptions or tentative values, while the second will be used for final results after the preliminary values have been either confirmed or modified. PROCEDURE IN TRANSFORMER DESIGN 155 SPECIFICATION Output ............ ' .................. i ,500 k.v.a. Number of phases ................. .... one H.T. voltage ......................... 88,000 L.T. voltage .......................... 6,000 Frequency ........................... 50 Maximum efficiency, to occur at full load and not to be less than .............. 98 . i% Voltage regulation, on 80 per cent power factor ............................. 5% Temperature rise after continuous full- load run ......................... . . 40 C. Test voltage: H.T. winding to case and L.T. coils .......................... 177,000 L.T. winding to case .................. 14,000 The calculated values of the various items are here brought together for reference and for convenience in following the successive steps in the design. The items are numbered to facilitate reference to the notes and more detailed calculations which follow. Items (i) and (2). L.T. Winding. By Formula (490), Art. 39, page 142, the volts per turn, for a shell-type power transformer, are 500,000 = 51.5, whence, r s = 66oo = I28 . 51-5 156 PRINCIPLES OF TRANSFORMER DESIGN DESIGN SHEET i. Volts per turn. L.T. WINDING (SECONDARY) Total number of turns Number of coils Number of turns per coil . Secondary current, amperes 6. Current density, amperes per sq. in. . . 7. Cross-section of each conductor, sq. in. 8. Insulation on wire, cotton tape, in. . . 9. Insulation between layers, in 10. Number of turns per layer, per coil. . . 11. Number of layers 12. Overall width of finished coil (say), in. 13. Thickness (or depth) of coil, with allowance for irregularities and bulging at center, in H.T. WINDING (PRIMARY) 14. Total number of turns 15. Number of coils 16. Number of turns per coil 17. Primary current, amperes 18. Current density, amperes per sq. in. . 19. Cross-section of each wire, sq. in. . . . 20. Insul. on wire (cotton covering), in. . . 21. Insul. between layers, fullerboard, in. . 22. Number of turns per layer, per coil. . . 23. Number of layers; in all but end coils 24. Overall width of finished coil, in 25. Thickness or depth of coil, in 26. Make sketch of assembly of coils, with necessary insulating spaces and oil ducts. Symbol. Assumed or Approxi- mate Values. 51-5 128 6 21.3 Final Values. 52.3 126 6 21 227 1575 1600 3strips, eacho. 16X0.3=0.144 | 0.026 2X0. 006) +o .012 = 0. 024 i 18 21 0.36 . 5 1680 18 80 in 2 coils; 95 in 16 coils 1 7 . 05 1640 0.04X0. 26 = 0.0104 2X0.008 = 0.016 O.OI2 I 95 0.31 6.75 PROCEDURE IN TRANSFORMER DESIGN 157 DESIGN SHEET Continued Symbol. Assumed or Approxi- mate Values. Final Values. 27. Size of " window " or opening for windings, in 12. 75X32 MAGNETIC CIRCUIT 28 Total flux (maxwells) $ 2.36Xio 7 29. Maximum value of flux density in core under windings (gausses) 30 Cross-section of iron under coils, sq. in. B 13,000 282 13,850 264 31 Number of oil ducts in core none 32. Width of oil ducts in core 33. Width of stampings under windings, in L ii II 34 Net length of iron in core in 24 35. Gross length of core, in 36. Cross-section of iron in magnetic cir- cuit outside windings sq in. . . S 27 264 37. Flux density in core outside windings (gausses) B 13,8^0 38. Average length of magnetic circuit under coils, in in. s 39. Average length of magnetic circuit outside coils in III 1 40 Weight of core, Ib , 82<;o 41 Losses in the iron, watts W\ II.OOO COPPER LOSSES 42 Mean length per turn of primary, ft. . IO 11 43. Resistance of primary. winding, ohms . 44. Full-load losses in primary (exc. cur- rent neglected) . . Ri 18.1 e-jgo 45. Mean length per turn of secondary, ft. IO. It 46. Resistance of secondary winding, ohms 47 Full-load losses in secondary watts R* 0.0962 4060 48. Total full-load copper losses, watts . . W c 10,220 . 158 PRINCIPLES OF TRANSFORMER DESIGN DESIGN SHEET Continued Symbol. Assumed or Approxi- mate Values. Final Values. 49. Total weight of copper in windings, Ib. 50. Efficiency at full load ' (unity power factor) 1700 o 08 5 51. Efficiency at other loads and power factors (refer to text following). 52. No-load primary exciting current, amperes . . Ic 2 I ^ REGULATION 53. Reactive voltage drop IiX v 2850 54. Equivalent ohmic voltage drop hRp 600 55. Regulation on unity power factor, per cent. . O 7^t? 56. Regulation on 80 per cent power fac- tor, per cent. 2 ^ DESIGN OF TANK, COOLING SURFACES 57. Effective cooling surface of tank, sq. in. 58. Number of watts dissipated from tank surface 19,360 46^0 18,860 59. Watts to be carried away by circula- ting water 17,470 60. Size and length of pipe in cooling coil,. 61. Approximate flow of water per minute, gallons it"X37o' 4. 37 62. Approximate weight of oil Ib 7 3OO 63. Estimated total weight of transformer, Ib. . 22,OOO (3) and (4). The number of separate coils is determined by the following considerations: (a) The voltage per coil should preferably not exceed 5000 volts. PROCEDURE IN TRANSFORMER DESIGN 159 (b) The thickness per coil should be small (usually within 1.5 in.) in order that the heat may readily be carried away by the oil or air in the ducts between coils (Refer Art. 23). (c) The number of coils must be large enough to admit of proper subdivision into sections of adjacent primary and secondary coils to satisfy the requirements of regu- lation by limiting the magnetic flux-linkages of the leak- age field. (d) An even number of L.T. coils is desirable in order to provide for a low- tension coil near the iron at each end of the stack. To satisfy (a), ^here'must be at least - - or, say- 18 H.T. coils. If an equal number of secondary coils were provided, we could, if desired, have as many as eighteen similar high-low sections which would be more than necessary to satisfy (c). The number of these high-low sections or groupings must be estimated now in order that the arrangement of the coils, and the num- ber of secondary coils, may be decided upon with a view to calculating the size of the " windows " in the mag- netic circuit. It is true that the calculations of reactive drop and regulation can only be made later; but these will check the correctness of the assumptions now made, and the coil grouping will have to be changed if neces- sary after the preliminary design has been carried some- what farther. The least space occupied by the insula- tion, and the shortest magnetic circuit, would be obtained by grouping all the primary coils in the center, with half the secondary winding at each end, thus giving only 160 PRINCIPLES OF TRANSFORMER DESIGN two high-low sections ; but this would lead to a very high leakage reactance, and regulation much worse than the specified 6 per cent. Experience suggests that about six high-low sections should suffice in a transformer of this size and voltage, and we shall try this by arranging the high-tension coils in groups of six, and providing six secondary coils (see Fig. 52). This gives us for item (4), J -f- = 2i.3 or, say, 21, whence T s = i26. Items (5) to (13). The secondary current is 7 S = I? ^ 00 ' 000 = 227 amperes. From Art. 20, we select 6600 A = 1600 as a reasonable value for the current density, .giving --=0.142 sq. in. for the cross-section of the IOOO secondary conductor. In order to decide upon a suitable width of copper in the secondary coils, it will be desirable to eti- mate the total space required for the windings so that the proportions of the " window " may be such as have been found satisfactory in practice. The space factor (Art. 41) is not likely to be better than o.i, which gives for the area of the "window" 2Xi26~Xo.i42 u . . , ., = 358 sq. in. Also, if a reasonable as- o.i sumption is that H = 2.5 times D (see Fig. 49, page 147), it follows that 2.5Z>X.D = 358; whence Z) = i2 inches. The clearance between copper and iron under oil, for a working pressure of 6600 volts (Formula (12), Art. 16), should be about 0.25+0.05X6.6 = 0.58 in. For the insulation between layers, we might have 0.02 in. for cotton, and a strip of 0.012 in. fullerboard, . PROCEDURE IN TRANSFORMER DESIGN 161 making a total of 21X0.032=0.67 in. The thickness of each secondary conductor will therefore be about 12 -(0.58+0.67+0.58) . , . , . -=0.485 in., which gives a width of ai ^ 2 =0.293 in. Let us make this 0.3- in., and 0.485' build up each conductor of three strips 0.16 in. thick, with 0.006 paper between wires (to reduce eddy cur- rent loss) and cotton tape outside. Allowing 0.026 in. for the cotton tape, and 0.012 in. for a strip of fuller- board between turns, the total thickness of insulation, measured across the layers, is 21 X (0.026 +0.024) = 1.05 in. A width of " window " of 12.75 m - ( see Fig. 52) will accommodate these coils. The current density- with this size of copper is Items (14) to (25). H.T. Winding. T p 66 = 1680. This may be divided into 16 coils of 95 turns each, and 2 coils of only 80 turns each, which would be placed at the ends of the winding and provided with extra insulation between the end turns (see Art. 14). According to Formula (13) of Ait. 16, the thickness of insulation consisting of partitions of fullerbcard with spaces between for oil circulation separating the H.T. copper from L.T. coils or grounded ircn, should not be less than 0.25+0.03X88 = 2 89 in. Let us make this clearance 3 in. Then, since the width of opening 162 PRINCIPLES OF TRANSFORMER DESIGN is 12.75 m " the maximum permissible depth of winding of the primary coils will be 12.75 6 = 6.75 in. The N . T 1,500,000 primary current (Item 17) is /*=-^- J = 17-05 amps. 00,000 (approx.). The cross-section of each wire is = 0.01065 sq. in. Allowing 0.016 in. for the total increase of thickness due to the cotton insulation, and 0.012 FIG. 52. Section through Windings and Insulation. in. for a strip of fullerboard between turns, the thick- ness of the copper strip (assuming flat strip to be used) must not exceed ( ) 0.028 = 0.043 in., which \ 95 / makes the width of copper strip equal to - ^ =0.248 0.043 in. Try copper strip 0.26X0.04=0.0104 sq. in., making A = 1640. PROCEDURES TRANSFORMER DESIGN 163 The two end coils, with fewer turns, would be built up to about the same depth as the other coils by putting increasing thicknesses of insulation between the end turns. Thus, since there is a total thickness of copper equal to 0.04 X (95 80) =0.6 in. to be replaced by insula- tion, we might gradually increase the thickness of fuller- board between the last eight turns from 0.012 in. to 0.15 in. Items (26) and (27). Size of Opening lor Windings. A drawing to a fairly large scale, showing the cross- section through the coils and insulation, should now be made. Oil ducts not less than \ in. or ^ in. wide should be provided near the coils to carry off the heat, and the large oil spaces between the H.T. coils and the L.T. coils and iron stampings, should be broken up by partitions of pressboard or other similar insulating material, as indicated roughly in a portion of the sketch, Fig. 52. In this manner the second dimension of the " window " is obtained. This is found to be 32 in., whence the copper space factor is (1680X0.0104) + (126 0<o_i445) _ 12.75X32 Items (28) to (41). The Magnetic Circuit. By Formula (i), Art. 2, 88,000 X io 8 4.44X50X1680 Before assuming a flux density for the core, let us calculate the permissible losses. 164 PRINCIPLES OF TRANSFORMER DESIGN The full load efficiency being 0.981, the total losses i, 500,000 X(i 0.081) are -^ ^- = 29,000 watts. Also, since 0.981 fW\ the ratio is approximately 0.925 (see Art. 39, under sub-heading The Factor b) y it follows that W c = -i = 1 5 , TOO watts. 1-925 whence W c = 29,000 15,100 = 13,900 watts. Let us assume the width of core under the windings (the dimension L of Fig. 49) to be 1 1 in. and the width, B, of the return circuit carrying half the flux, to be 5.5 in. Then the average lengtfi of the magnetic circuit, measured along the flux lines, will be 2(12.75 + 5.5+32 + 5-5) = 111-5 in - If the flux density is taken at 13,000 gausses (selected from the approximate values of Art. 20) the cross- 2 section of the iron is ^ = 282 sq. in. The 13,000X6.45 watts lost per pound (from Fig. 27) are w = 1.27, whence the total iron loss is , */' 'h ^ *~* 1^ = 1.27X0.28X282X111.5 = 11,200 watts, which is considerably less than the permissible loss. It is not advisable to use flux densities much in excess of the selected value of 13,000 gausses for the following reasons: (a) The distortion of wave shapes when the mag- netization is carried beyond the " knee " of the B-H curve. PROCEDURE IN TRANSFORMER DESIGN 165 (b) The large value of the exciting current. (c) The difficulty of getting rid of the heat from the surface of the iron when the watts lost per unit volume are considerable. Let us, therefore, proceed with the design on the basis of 14,000 gausses as an upper limit for the flux density. If no oil ducts are provided between sections of the stampings, the stacking factor will be about 0.89. A gross length of 27 in. (Item 35) gives 24 in. for the net length, and a cioss-section of 24 X 1 1 = 264 sq. in. Whence $ = 13,850 gausses, and the total weight of iron is 264X111. 5X0.28 = 8250 Ib. The watts per pound, from Fig. 27, are ^=1.44, whence Wi = 1 1 ,900. Items (42) to (49), Copper Loss. The mean length per turn of the windings is best obtained by making a draw- ing such as Fig. 53. This sketch shows a section through the stampings parallel with the plane of the coils. The mean length per turn of the secondary, as measured off the drawing, is 122 in., and since the length per turn of the primary coils will be about the same, this dimen- sion will be used in both cases. Taking the resistivity of the copper at 0.9X10" ohms per inch cube (see The Factor k c , in Art. 39), the primary resistance (hot) is D 0.9X122X1680 /Ci = - - = 18.1 ohms, i o 6 X 0.0104 whence the losses (Item 44) are (i7.o5) 2 X 18.1 = 5260 watts. 166 PRINCIPLES OF TRANSFORMER DESIGN For the secondary winding we have 0.9X122X126 R2 = -=0.0062 ohm. io 6 X 0.144 12%- Zl FIG. 53. Section through Coil and Stampings. whence the losses (Item 47) are (227)2 Xo.og62 =4960 watts, and ^ = 5360+4960=10,220 watts, which is appreciably less than the permissible copper loss. PROCEDURE IN TRANSFORMER DESIGN 167 It is at this stage of the calculations that changes should be made, if desirable, to reduce the cost of mate- rials, by making such modifications as would bring the losses near to the permissible upper limit. The obvious thing to do in this case would consist in increasing the current density in the windings, and perhaps making a small reduction in the number of turns. A considerable saving of copper would thus be effected without neces- sarily involving any appreciable increase in the weight of the iron stampings. Since this example is being worked through merely for the purpose of illustrating the manner in which fundamental principles of design may be applied in practice, no changes will be made here to the dimensions and quantities already calculated. The weight of copper (Item 49) is 0.32 (122X1680X0.0104) + 0.32(122X126X0.144) = 1,700 Ib. Items (50) and (51). Efficiency. The full-load effi- ciency on unity power factor is =0.985. 1,500,000+11,900+10,220 The calculated efficiencies at other loads are: At ij full load 0.985 At f full load 0.984 At | full load 0.981 At | full load 0.968 168 PRINCIPLES OF TRANSFORMER DESIGN The full-load efficiency on 80 per cent power factor is 1,500,000X0.8 (1,500,000X0.8)4-22,120 = 0.982. Item (52). Open-circuit Exciting Current. Using the curves of Fig. 45 (see Art. 35 for explanation), we obtain for a density = 13,850 the value 23 volt-amperes per pound of core. The weight of iron (Item 40) being 8250 lb., it follows that the exciting current is r 8250X23 - This is 12.6 per cent of the load component, which is rather more than it should be. If the design is altered, as previously suggested, to reduce the amount of copper, this will result in a reduction of the opening in the iron, and, therefore, also of the length of the magnetic circuit. It is, however, clear that the flux density (Item 29) must not be higher than 13,850 gausses. If the design were modified, it might be advisable to reduce this value by slightly increasing the cross-section of the magnetic cirduit. The fact that the exciting current component is fairly large relatively to the load current will lead to a small increase in the calculated copper loss (Item 44); but for practical purposes it is unnecessary to make the correction. Items (53) to (56) Regulation. Referring to Fig. 52, it is seen that there are six high-low sections, all about equal, since the smaller number of turns in two out of eighteen primary coils is not worth considering in calcu- PROCEDURE IN TRANSFORMER DESIGN 169 lations which cannot in any case be expected to yield very accurate results. The quantities for use in Formula (40) of Art. 34 have, therefore, the following values: Ti= :L * s -=2*o', 71 = 17.05; 7 = 10.15X12X2.54 = 310 cm.; # = 3X2.54 = 7.62 cm.; # = 1.7X2.54 = 4.32 cm.; 5 = 0.38X2.54 = 0.965 cm.; h= 12.75X2.54 = 32.4 cm. whence the induced volts per section are, 7iXi=475 volts. Since there are six sections, and all the turns are in series, the total reactive drop at full load is IiXp = 475 X 6 = 2850 volts, which is only 3 . 24 per cent of the primary impressed voltage. By Formula (35) Art. 33, the equivalent primary resistance is RJ, = I&.I + ( -*- } X 0.0062 = 3 5. 2 ohms; \ I2 6/ whence IiRp = 600 volts. which is 0.683 per cent of the primary impressed voltage. By Formula (47), Art. 36, when the power factor is unity (cos = i). Regulation = 0.683 +0 = 0.683 per cent 170 PRINCIPLES OF TRANSFORMER DESIGN The more correct value, as obtained from Formula (46) is 0.735. When the power factor of the load is 80 per cent, the approximate formula which is quite sufficiently accu- rate in this case gives Regulation = (0.683 X 0.8) + (3 . 24 X 0.6) = 2.5 per cent (approx.) on 80 per cent power factor. This is very low, and considerably less than the specified limit of 5 per cent. It is possible that the specified reg- ulation might be obtained with only 4, instead of 6, high- low groups of coils, and in order to produce the cheapest transformer to satisfy the specification, the designer would have to abandon this preliminary design until he had satisfied himself whether or not an alternative design with a different grouping of coils would fulfill the requirements. It is clear from the inspection of Fig. 52 that an arrangement with only four L.T. coils and (say) sixteen H.T. coils would considerably reduce the size of the opening in the stampings, thus saving materials and, incidentally, reducing the magnetizing current, which is abnormally high in this preliminary design. Items (57) to (61). Requirements for Limiting Tem- perature Rise. A plan view of the assembled stampings should be drawn, as in Fig. 54, from which the size of containing tank may be obtained. In this instance it is seen that a tank of circular section 5 ft. 3 in. diam- eter will accommodate the transformer. The heiglit of the tank (see Fig. 55) will now have to be estimated in order to calculate the approximate cooling surface. This height will be about 90 in., and if we assume a PROCEDURE IN TRANSFORMER DESIGN 171 smooth surface (no corrugations), the watts that can be dissipated continuously are =465; FIG. 54. Assembled Stampings in Tank of Circular Section. 4 the multiplier 0.34 being obtained from the curve, Fig. 32 of Art. 25. The watts to be carried away by the circulating water are (10,220+11,900) 4650= 17,470. From data given in Art. 29, it follows that a coil made of i| in. 172 PRINCIPLES OF TRANSFORMER DESIGN tube should have a length of 1 7>47 = 270 ft. I2XI.25X7T r H. T. Terminal as ) detailed in Fig. FIG. 55. Sketch of isoo-k.v.a., 88,ooo-volt Transformer in Tank. Assuming the coil to have' an average diameter of 4 ft. 8 in., the number of turns required will be about 25. PROCEDURE IN TRANSFORMER DESIGN 173 On the basis of \ gal. of water per kilowatt, the required rate of flow for an average temperature dif- ference of 15 C. between outgoing and ingoing water is 0.25X17.47=4.37 gal. per minute. This amount may have to be increased unless the pipes are kept clean and free from scale. The . completed sketch, Fig. 55, indicates that a tank 87 in. high will accommodate the transformer and cooling coils, and the corrected cooling surface for use in temperature calculations (see Art. 25) is therefore 18,860 sq. in. This new value for Item 57 has been put in the last column of the design sheet; but the items immediately following, which are dependent upon it, have not been corrected because the difference is of no practical im- portance. Hottest Spot Temperature. The manner in which the temperature at the center of the coils may be calculated when the surface temperature is known, was explained in Art. 23. It is unnecessary to make the calculation in this instance because the coils are narrow and built up of flat copper strip. There will be no local " hot spots " if adequate ducts for oil circulation are provided around the coils. Items (62) and (63). Weight of Oil and of Complete Transformer. The weight of an average quality of transformer oil is 53 Ib. per cubic foot, from which the total weight of oil is found to be about 7300 Ib. The 174 PRINCIPLES OF TRANSFORMER DESIGN calculated weights of copper in the windings (Item 49) and iron in the core (Item 40) are 1700 Ib. and 8250 lb., respectively. The sum of these three figures is 17,250 lb. This, together with an estimated total of 4750 lb. to cover the tank; base and cover, cooling coil, terminals, solid insulation, framework, bolts, and sundries, brings the weight of the finished transformer up to 22,000 lb. (in- 22 OOO eluding oil) ; or - - = 14.65 lb. per k.v.a. of rated full- 1500 load output. Several details of construction have not been referred to. It is possible, for instance, that tappings should be provided for adjustment of secondary voltage to com- pensate for loss of pressure in a long transmission line. These should preferably be provided in a portion of the winding which is always nearly at ground potential. It is not uncommon to provide for a total voltage varia- tion of 10 per cent in four or five steps, which is accom- plished by cutting in or out a corresponding number of turns, either on the primary or secondary side, which- ever may be the most convenient. Mechanical Stresses in Coils. The manner in which the projecting ends of flat coils in a shell- type transformer should be clamped together is shown in Fig. 16 of Art. 9. Let us calculate the approximate pressure tending to force the projecting portion of the secondary end coils outward when a dead short-circuit occurs on the trans- former. The force in pounds, according to Formula (4), is JL IfL max Jj&m. 8,896,000* PROCEDURE IN TRANSFORMER DESIGN 175 For the quantities T and /, we have and /, being the average length of the portion of a turn projecting beyond the stampings at one end, is 7 10.15X12 / = -- 27=34 in. or 86 cms. The value of the quantities 7 max and B &m depends on the impedance of the transformer. With normal full- load current, the impedance drop is volts, where the quantities under the radical are the items 53 and 54 of the design sheet. In order to choke back the full impressed voltage, the current would have to be about = (say) thirty times the normal full-load value. 2QIC Thus the current value for use in Formula (4) , on the sine wave assumption, will be /max = 30 X 2 2 7 X "\/2 = 9650 amperes. The density of the leakage flux through the coil is less easily calculated; but, since the reactive voltage was calculated on the assumption of flux lines all parallel to the plane of the coil, we may now consider a path one square centimeter in cross-section and of length equal to 176 PRINCIPLES OF TRANSFORMER DESIGN the depth of the coil (about 29 cms.) in which the leakage flux will have the average value. -Bam = - X 2 1 X 9650 X = 4400 gausses, 2LIO 29J whence, by Formula (4), Force in Ib. = ^ <X^5X_44o = g6 ft 8,896,000 This is the force F of Fig. 16, distributed over the whole of the exposed surface of the end coil. An equal force will tend to deflect outward the secondary coil at the other end of the stack. If an arrangement of straps with two bolts is adopted as shown in Fig. 16 each bolt must be able to withstand a maximum load of 4370 Ib. Bolts f in. diameter will, therefore, be more than suf- ficient to prevent displacement of the coils, even on a dead short circuit. CHAPTER VI TRANSFORMERS FOR SPECIAL PURPOSES 44. General Remarks, When applying the funda- mental principles of electrical design to special types of apparatus, it is necessary to consider what are the chief characteristics of such apparatus and wherein they differ from those of the more usual types. The apparatus dealt with in the preceding chapters is the potential transformer for use either, as large units, in power stations, or in smaller sizes, as means of distributing electric power in residential or industrial districts. A few special types of transformer will now be considered; but the treat- ment will be brief, with the object of avoiding useless repetitions. Attention will be given mainly to such dis- tinctive features or peculiarities as may have an impor- tant bearing on the design. 45. Transformers for Large Currents and Low Volt- ages. Electric furnaces are built to take currents up to 35,000 amperes at about 80 volts usually three-phase. Welding transformers must give large currents at a com- paratively low voltage. A current of 2000 amperes at 5 volts would probably be required for rail welding on an electric railroad. Transformers for thawing out frozen water pipes need not necessarily be specially designed because standard distributing transformers 177 178 PRINCIPLES OF TRANSFORMER DESIGN connected to give about 50 volts are used successfully for this purpose. A transformer of 12 k.v.a. normal rating, capable of giving up to 600 amperes with a max- imum pressure of 30 volts for short periods of time in cold weather, will probably answer all requirement for the thawing of house service pipes up to i| in. diameter. A current of 400 amperes will thaw out a i-in. pipe in about half an hour. In the design of all transformers for large currents, especially when they are liable to be practically short- circuited, the leakage reactance (see Art. 34) is a matter of importance. The permissible maximum current on a short circuit should be specified. In some cases, sepa- rate adjustable reactance coils (usually on the high- voltage side) are provided for the purpose of regulating the current from transformers used for welding and sim- ilar processes. Another point to be watched in the design of trans- formers for large currents is the eddy current loss in the copper (see Art. 20), which must be minimized by prop- erly arranging and laminating the secondary winding and leads. The mechanical details in the design of secondary terminals and leads also require careful atten- tion. 46. Constant-current Transformers. Circuits with incandescent or arc lamps connected in series require the amount of current to be approximately constant regard- less of the number of lamps on the circuit. If it is de- sired to supply series circuits of this nature from constant potential mains, special transformers are required, so designed as to give variable voltage at the secondary TRANSFORMERS FOR SPECIAL PURPOSES 179 terminals, with a constant voltage across the primary terminals. The variations in the secondary voltage are automatic, being the result of very small changes in the secondary current, brought about by switching lamps in or out of the circuit. In other words, the secondary voltage must follow as nearly as possible the variations in the impedance of the external circuit, so that a doubled impedance would very nearly bring about a doubling of the secondary voltage, the drop in current being as small as possible. Automatic regulation of this kind may be obtained by means of an ordinary transformer having a large amount of magnetic leakage, as for instance a core type trans- former purposely constructed with the primary turns on one limb and the secondary turns on the other limb, as shown diagrammatically in Fig. i of Art. 2. The vector diagram of such a transformer has been drawn in Fig. 56, based on the simplified diagram, Fig. 48 (Art. 36), which should be consulted for the meaning of the vectors. The same notation has been used in Fig. 56 as in Fig. 48, and it is to be observed that, on account of the leakage flux being a large percentage of the useful flux, a small reduction in the current, from I\ to 7'i, will automatically cause the vector E e (which is a measure of the secondary voltage) to become E e f , just twice as great. Although by suitably designing a transformer with considerable leakage flux, a small reduction in the reactive drop (the vector I\X P of Fig. '56) will produce a large increase in the secondary voltage, it is obvious that still better results would be obtained if the reactance (or amount of leakage flux) could be made to decrease at a 180 PRINCIPLES OF TRANSFORMER DESIGN greater rate than the current. Thus, if an increase of current could be made to bring about a change in the permeance of the leakage paths, the reactive drop, instead of being proportional to the current, might be made to increase at a greater rate than the current, FIG. 56. Vector Diagram of Transformer with Large Amount of Leak- age Flux. and so bring about the condition illustrated by Fig. 57 where the same result, i.e., a doubling of the secondary voltage is seen to be brought about by a very much smaller reduction in the amount of the current. It is evident that the primary volt-amperes must remain practically constant at all loads, and the fact that TRANSFORMERS FOR SPECIAL PURPOSES 181 the actual secondary output may vary considerably with changes in the resistance of the external circuit, is accounted for by the alteration in the power factor of the primary circuit. Thus, since the input and output of a transformer must be the same except for the internal losses, the changes of input with an almost constant Sill FIG. 57. Vector Diagram of Transformer with Variable Leakage Reactance. Ej>Ip product are accounted for by the changes in the angle </> of Fig. 57. Fig. 58 illustrates the principle of construction of the constant-current transformer with variable magnetic leakage. One coil is stationary while the other is movable, being suspended from a pivoted arm provided with a counterweight, and free to slide up and down on the cen- 182 PRINCIPLES OF TRANSFORMER DESIGN tral core of a shell-type magnetic circuit. The movable coil may be either the primary or the secondary, and by careful adjustment of the balance weight, a very small jchange in the current may be made to produce a con- siderable change in the relative position of the coils, thus greatly altering the relation between the- leakage and FIG. 58. Constant Current Transformer with " Floating " Coil. useful flux components, the (vectorial) sum of which passing through the primary coil must always remain practically constant. With the two coils in contact, the maximum secondary voltage corresponding to the maximum number of lamps in series is obtained; while on short-circuit the movable coil will be pushed as far away from the sta- TRANSFORMERS FOR SPECIAL PURPOSES 183 tionary coil as the construction of the transformer will admit. Except for the difficulty of calculating accu- rately the amount of the leakage flux-linkages corre- sponding to these two conditions, the design of a constant-current transformer for any given output is a simple matter. Regulation is not usually required over a range greater than from full load to about one- third of full load, and this can be obtained with a cur- rent variation not exceeding i per cent. The force tending to move the coils apart can readily be calculated with the aid of Formula (4) Art. 9; but since the quantity B am cannot be predetermined with great accuracy except in the case of standard designs for which data have been accumulated final adjustments must be made after completion, by the proper setting of the counterweight. Constant-current transformers for arc-lamp circuits off constant pressure mains require a secondary current between 6.5 and 10 amperes, and they usually operate in conjunction with a mercury arc rectifier to change the alternating current into a continuous current. Trans- formers for small outputs may be air cooled, while the larger units should be oil-immersed and, if necessary, cooled by circulating water. The full-load efficiency of constant-current trans- formers with movable coils for use on 2200-volt circuits ranges from 90 per cent for 3 kw. output on 6o-cycle circuits to 96 per cent for 30 kw. output on 25-cycle circuits. 47. Current Transformers for Use with Measuring Instruments. These transformers are of comparatively 184 PRINCIPLES OF TRANSFORMER DESIGN small size, their chief function being to provide a current for measuring-instruments which shall be as nearly as possible proportional to the line current passing through the primary coils. By their use it is possible to trans- form very large currents to a current of a few amperes which may conveniently be carried to instruments of standard construction mounted on the switchboard panels or in any convenient position preferably not very far removed from the primary circuit. Again, in the case of high-potential circuits, even if the reduction of current is not of great importance, the fact that the sec- ondary circuit of the current transformer can be at ground potential renders unnecessary the special instru- ments and costly methods of insulation that would be required if the line current of high- voltage systems were taken through the measuring instruments. A current transformer does not differ fundamentally from a potential transformer; but since the primary coils are in series with the primary circuit, the voltage across the terminals will depend upon the induced volts, which, in their turn, depend upon the impe- dance of the secondary circuit. With the secondary short-circuited, the voltage absorbed will be a mini- mum, and the input of the transformer will be approxi- mately equal to the copper losses, because a very small amount of flux will then be sufficient to generate the required voltage, and the iron losses will be negligible. The vector diagram for a series transformer does not differ from that of a shunt transformer, but Figs. 59 and 60 have been drawn to show clearly the influence of the magnetizing current on the relation between TRANSFORMERS FOR SPECIAL PURPOSES 185 the total primary and secondary currents. Fig. 59 shows the- vector relations when the power factor is unity, while in Fig. 60 there is an appreciable lag be- tween the current and e'.m.f. in the secondary circuit. When a current transformer is used in connection with an ammeter only, the essential condition to be fulfilled is that the ratio /./ 1 1 7\ or T * p \ *p be as nearly constant as possible over the whole range of current values. When the secondary current is passed through lw FIG. 59. the series coil of a wattmeter, it is equally important that L be as nearly as possible opposite in phase to IP, or, in other words, that the angle I P OIi be very small. A diagram, such as Fig. 60, may be constructed for any given condition of load, the amount of the flux B and therefore the exciting current I e being de- pendent upon the impedance of the secondary circuit, since this determines the necessary secondary voltage. On the sine wave assumption, it is an easy matter to express the quantity 01 p in terms of the secondary 186 PRINCIPLES OF TRANSFORMER DESIGN current and the two components of the exciting current. The vector 01 \ is a measure of the secondary current, IT \ being simply I A - ), and it is easily seen that M vl IP = \/(/i sin 0+/ ) 2 + (/i cos d+I u y, whence the ratio can be calculated for any power 1 v factor (cos 0) and any values of the secondary current D C FIG. 60. and voltage. It is interesting to note that, on a load of unity power factor (cos 0=i), the magnetizing com- ponent of the total exciting current does not appre- ciably affect the relation between the magnitudes of the primary and secondary currents, and for all practical TRANSFORMERS FOR SPECIAL PURPOSES 187 purposes the difference, under this particular load con- dition, is ,.,.,. iron loss (watts) L-D /I =-/,= e.m.f . induced in primary (volts) ' If this difference were always proportional to the primary current, there would be no particular advantage in keeping it very small; but since the power factor is not always unity, and variations in current mag- nitudes may be brought about by phase differences, it is always advisable to aim at obtaining an exciting current which shall be a very small percentage of the total primary current. The phase difference between I p and I\ (see Figs. 59 and 60) may be expressed as This angle must be very small, especially when the transformer is for use with a wattmeter. It should never exceed i minute, and should preferably be within thirty seconds. This condition can only be satisfied, with varying values of 6, by making the exciting current (especially the magnetizing component 7 ) very small relatively to the main current. It is therefore neces- sary to use low flux densities in ^the cores of series trans- formers for use with instruments, and this incidentally leads to small core losses and a small " energy " com- ponent (Iw) of the total exciting current. Flux densities ranging from 1500 to 2500 gausses at full load are not 188 PRINCIPLES OF TRANSFORMER DESIGN uncommon in well-designed series instrument trans- formers. Fig. 6 1 gives approximate losses per pound of transformer iron for these low densities which are not included in the curves of Fig. 27. Although curves for alloyed steel are not given, the losses may be approximately estimated by referring to Fig. 27 (Art. 20) and noting the relative positions of the curves for the two qualities of material. When the primary current is large, a convenient form of current transformer is one with a single turn of primary, that is to say, a straight bar or cable passing through the opening in the iron core. This is quite satisfactory for currents of 1000 amperes and upward, and the construction is t permissible with currents as low as 300 amperes, especially when the transformer is to be used 'in connection with a single ammeter, i.e., without a wattmeter, or second instrument, or relay coil, in series. The designer should, however, aim to get 1000 to 1500 ampere turns, or more, in each winding of a series instrument transformer. Although the presence of the exciting current com- ponent of an iron-cored transformer renders a constant ratio of current transformation theoretically unattain- able over the whole range of current values, this does not mean that any desired ratio of transformation cannot be obtained for a particular value of the primary current. It is, of* course, a simple matter to eliminate the error due to the presence of the exciting current (T \ -] that any Is/ desired current transformation may be obtained for a TRANSFORMERS FOR SPECIAL PURPOSES 189 .<y / 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Total watts per pound sw FIG. 61. Losses in Transformer Iron at Low Flux Densities. 190 PRINCIPLES OF TRANSFORMER DESIGN specified value of the primary current. If the ratio of transformation is correct at full load, it will be prac- tically correct over the range from f to full-load cur- rent, the error being most noticeable with the smaller values of the main current. The following figures are typical of the manner in which the transformation ratio of series instrument transformers is likely to vary. Percentage of Full-load Current. PERCENTAGE DEPARTURE FROM FULL-LOAD RATIO. A B C IOO o 0.25 I .0 3-0 6.0 O 0.04 o. 16 o-5 i .0 o o.S 2.O 6.0 12.0 7C cro 2S IO Column A gives average values: column B shows how small the error may be in well-designed transformers for use with wattmeters or other instruments demand- ing constancy in the current ratios; while C refers to commercial current transformers for use with relays, trip coils of switches, and other apparatus which does not call for great accuracy in the transforming ratio. In all cases a fairly low power factor is assumed, and a rated full-load output of about 50 volt-amperes. If the same transformers were to operate on an external circuit of reduced resistance and unity power factor, the percentage error would be considerably smaller. No special features other than reliability of insulation, and freedom from overheating have to be considered TRANSFORMERS FOR SPECIAL PURPOSES 191 in connection with series transformers used for oper- ating regulating devices or protective apparatus such as trip coils on automatic overload circuit-breakers. The flux density in the core may then be higher than in instrument transformers. 48. Auto-transformers. An ordinary transformer be- comes an auto-transformer, or compensator, when the FIG. 62. Ordinary Transformer Connected as Auto-transformer. connections are made as in Fig. 62. One terminal is then common to both circuits, the supply voltage being across all the turns of both windings in series, while the secondary or load voltage is taken off a por- tion only of the total number of turns. This arrange- ment would be adopted for stepping down the voltage; but by interchanging the connections from the supply 192 PRINCIPLES OF TRANSFORMER DESIGN circuit and the load, the auto- transformer can be used equally well for stepping up the voltage. There is little advantage to be gained by using auto- transformers when the ratio of transformation is large; but for small percentage differences between the supply and load voltages, considerable economy is effected by using a,n auto-transformer in place of the usual type with two distinct windings. Let 7\, = the number of turns between terminals a and c (Fig. 62) ; T s = the number of turns between terminals c and b; then (T p +Ts) = ihe number of turns between terminals a and b. The meaning of other symbols is indicated on Fig. 62. The ratio of transformation is E v p s ( x = - =r (54) If used as an ordinary transformer, the transforming ratio would be -r-i (55) J- s The ratio of currents is -^-r, (56) TRANSFORMERS FOR SPECIAL PURPOSES 193 while the current I c in the portion of the winding com- mon to both primary and secondary is obtained from the equation 1 cJ- s 1 pi p, whence / = /,('-!), ..... (57) or, in terms of the secondary current, (58) None of the above expressions takes account of the exciting current and internal losses. The volt-ampere output, as an auto-transformer, is E s ls'y but part of the energy passes directly from the primary into the secondary circuit. For the purpose of determining the size of an auto-transformer, we require to know its equivalent transformer rating. The volt-amperes actually transformed are E S I C , whence Output as ordinary transformer _Ic_r i . ^ Output as auto- transformer I s r which shows clearly that it is only when the ratio of voltage transformation (r) is small that an appreciable saving in cost can be effected by using an auto-trans- former. The ratio of turns, and the amount of the currents to be carried by the two portions of the winding having been determined by means of the preceding formulas, the design may be carried out exactly as for an ordi- 194 PRINCIPLES OF TRANSFORMER DESIGN nary potential transformer, attention being paid to the voltage to ground, which may not be the same in the auto-transformer as in an ordinary transformer for use under the same conditions. Auto-transformers are, however, rarely used on high voltage circuits, although there appears to be no objection to their use on grounded systems. Effect of the Exciting Current in Auto-transformers. In the foregoing discussions, the effect of the exciting current was considered negligible. Tin's assumption is > p Tp turns Tg turns h)CR5oW5 NJOtffl(H)0000000(ft)&0(r I* SAAAAAWW\WAA/W\W^A/^ li Load FIG. 63. Diagram of Connections of Auto- transformer. usually permissible in practice; but since it may some- times be necessary to investigate the effect of the exciting current components, a means of drawing the vector diagram showing the correct relation of the current components will now be explained. Fig. 63 is similar to Fig. 62 except that it shows the connections in a simplified manner. The arrows indicate what we shall consider the positive directions of the various currents. TRANSFORMERS FOR SPECIAL PURPOSES 195 The fundamental condition to be satisfied is that the (vectorial) addition of all currents flowing to or from the junction c or b shall be zero. Whence, Ip + Is = Ic ...... (60) Let L stand for the exciting current when there is no current flowing in the secondary circuit. This is readily calculated exactly as for an ordinary trans- former with Ep volts across (T P + T S ) turns of winding. Then, since the resultant exciting ampere turns must always be approximately (Tp-\-T s )I e , the condition to be satisfied under load is V, (6i) which, if we divide by T s , becomes If I c in this equation is replaced by its equivalent value in terms of the other current components, as given by Equation (60), we get j s -- / f rl p = rr e -l s (63) The vector diagram Fig. 64 satisfies these conditions; the construction being as follows: Draw OB and OE S to represent respectively the phase of the magnetic flux and induced voltage. Draw OI to represent the current in the secondary circuit in its proper phase relation to E s . Now calculate the 196 PRINCIPLES OF TRANSFORMER DESIGN exciting current I e on the assumption that it flows through all the turns (T P +T S ], and draw OM, equal to rI C j in its proper phase relation to OB. Join ML and determine the point C by making CI S = - . Then, since I S M is the vectorial difference between rl e and I s , it follows from Equation (63) that it is equal to *E. FIG. 64. Vector Diagram of Auto-transformer, Taking Account of Exciting Current. rI P , whence CI S =I V , and CM = (ri)I p . Also, since OC is the vectorial sum of I s and I p , it follows from Equation (60) that OC is the vector of the current I c in the portion of the winding common to both circuits. In this manner the correct value and phase relations of the currents ID and 7 C , in the sections ac and cb of the winding, can be calculated for any given load conditions. TRANSFORMERS FOR SPECIAL PURPOSES 197 49. Induction Regulators. In order to obtain a vari- able ratio of voltage transformation, it is necessary either to alter the ratio of turns by cutting in or out sections of one of the windings, or to alter the effective flux- linkages by causing more or less of the total flux linking with the primary to link with the secondary. The principle of variable ratio transformers of the moving iron type is illustrated by the section shown in Fig. 65. This is a diagrammatic representation of a single-phase induction regulator with the primary coils on a cylindrical iron core capable of rotation through an angle of go degrees. The secondary coils are in slots in the stationary portion of the iron cir- cuit. The dotted lines show the general direction of the magnetic flux when the primary is in the position corresponding to maximum secondary voltage. As the movable core is rotated either to the right or left, the secondary voltage will decrease until, when the axis AB occupies the position CD, the flux lines linking with the secondary generate equal but opposite e.m.f.s in symmetrically placed secondary coils, with the result that the secondary terminal voltage falls to zero. If current is flowing through the secondary winding as will be the case when the transformer is connected up as a " booster " or feeder regulator the reactive voltage due to flux lines set up by the secondary current and passing through the movable core in the general direction CD, will be considerable unless a short-circuited winding of about the same cross-section as these cond- ary is provided as indicated in Fig. 65. It is immaterial whether the winding on the movable 198 PRINCIPLES OF TRANSFORMER DESIGN core be the primary or secondary ; but if the primary is on the stationary ring, the short-circuited coils must also be on the ring. The chief difficulty in the design of induction regulators FIG. 65. Diagram of Single-phase Variable-ratio Transformer of the Moving-iron Type. arises from the introduction of necessary clearance gaps in the magnetic circuit, and the impossibility of arranging the coils as satisfactorily as in an ordinary static transformer so as to avoid excessive magnetic TRANSFORMERS FOR SPECIAL PURPOSES 199 leakage. A large exciting current component and an appreciable reactive voltage drop are characteristic of the induction voltage-regulator. Fig. 66 is a diagram showing a single-phase regu- lating transformer of the type illustrated in Fig. 65 connected as a feeder regulator, the secondary being in series with one of the cables leaving a generating station to supply an outlying district. The movement of the iron core can be accomplished either by hand, or auto- matically by means of a small motor which is made to rotate in either direction through a simple device actuated by potential coils or relays. The lower diagram of Fig. 66 shows the core carrying the primary winding in the position which brings the voltage generated in the ring winding to zero. The flux lines shown in the diagram are those produced by the magnetizing current in the primary winding; but there are other flux lines not shown in the diagram which are due to the current in the ring winding. It is true that the movable core carries a short-cir- cuited winding not shown in Fig. 66 which greatly reduces the amount of this secondary leakage flux; but it will nevertheless be considerable, and the secondary reactive voltage drop is likely to be excessive, especially if the ring winding consists of a large number of turns. An improvement suggested by the writer at the time * when this type of apparatus was in the early stages of its development, consists in putting approximately half the secondary winding on the portion of the magnetic circuit which carries the primary winding, the balance * The year 1895. 200 PRINCIPLES OF TRANSFORMER DESIGN FIG- 66. Variable-ratio Transformer Connected as Feeder Regulator. TRANSFORMERS FOR SPECIAL PURPOSES 201 of the secondary turns being put on the other portion of the magnetic circuit. The connections are made as in Fig. 67, the result being that the movement of the rotating core, to produce the full range of secondary voltage from zero to the desired maximum, is now 180 instead of 90 as in Fig. 66; but since, under the same conditions of operation, the ring winding for a given section of iron will carry only half the number of turns that would be necessary with the ordinary type (Fig. 66), the secondary reactive voltage drop is very nearly halved. This is one of the special features of the regulating transformers manufactured by Messrs. Switchgear & Cowans, Ltd., of Manchester, England. Consider the case of a single-phase system with 2200 volts on the bus bars in the generating station. The voltage drop in a long outgoing feeder may be such as to require the addition of 200 volts at full load in order to maintain the proper pressure at the distant end. If this feeder carries 100 amperes at full load, the neces- sary capacity of a boosting transformer of the type shown diagrammatically in Fig. 67 is 20 k.v.a. This variable-ratio transformer, with its primary across the 2 200- volt supply, and its secondary in series with the outgoing feeder, will be capable of adding any voltage between o and 200 to the bus-bar voltage. As an alternative, the supply voltage at the generating station end of this feeder may be permanently raised to 2300 volts by providing a fixed-ratio static transformer external to the variable-ratio induction regulator and connected with its secondary in series with the feeder. An induction regulator of the ordinary type (Fig. 66) 202 PRINCIPLES OF TRANSFORMER DESIGN Position of Zero Secondary Pressure Position of Maximum Secondary Pressure FIG. 67. Moving-iron Type of Feeder Regulator with Specially Drranged Secondary Winding. TRANSFORMERS FOR SPECIAL PURPOSES 203 capable of both increasing and decreasing the pressure by 100 volts, will then provide the desired regulation between 2200 and 2400 volts. The equivalent trans- 100X100 former output of this regulator will be = 10 1000 k.v.a. The Polyphase Induction Regulator. Two or three single-phase regulators of the type illustrated in Fig. 65 may be used for the regulation, of three-phase circuits; but a three-phase regulator is generally preferable. The three-phase regulator of the inductor type is essentially a polyphase motor with coil-wound not squirrel-cage rotor, which is not free, to rotate, but can be moved through the required angle by mechanical gearing oper- ated in the same manner as the single-phase regulator. The rotating field due to the currents in the stator coils induces in the rotor coils e.m.f.'s of which the magnitude is constant, since it depends upon the ratio of turns, but of which the phase relation to the primary e.m.f. depends upon the position of the rotor coils relatively to the stator coils. When connected as a voltage regulator for a three-phase feeder, the vectorial sum of the secondary and primary volts of a three-phase induction regulator will depend upon the angular dis- placement of the secondary coils relatively to the cor- responding primary coils. Mr. G. H. Eardley-Wilmot * has pointed out certain advantages resulting from the use of two three-phase induction regulators with secondaries connected in series, for the regulation of a three-phase feeder. By making * The Electrician, Feb. 19, 1915, Vol. 74, page 660, 204 PRINCIPLES OF TRANSFORMER DESIGN the connections so that the magnetic fields in the two regulators rotate in opposite directions, the resultant secondary voltage will be in phase with the primary voltage. The torque of one regulator can be made to balance that of the other, thus greatly reducing the power necessary to operate the controlling mechanism. INDEX A PAGE Absolute unit of current 26 Air-blast, cooling by, 88 All-day efficiency (see Efficiency). Alloyed-iron transformer stampings 19 Ampere-turns to overcome reluctance of joints 127 Analogy between dielectric, and magnetic, circuits 33 Auto-transformers 191 B B-H curves (see Magnetization curves). Bracing transformer coils (see Stresses in transformer coils). Bushings (see Terminals). Calorie, definition 99 Capacity current 41 electrostatic 33, 36 of plate condenser 40 Capacities in series 42 Charging current (Capacity . current) 41 Classification of transformers 14 Compensators 191 Condensers in series 42 Condenser type of bushing , 62 Conductivity, heat 80, 82, 87 Constant-current transformers 178 Construction of transformers 17, 24, 31 205 206 INDEX PAGE Cooling of transformers 14, 88, 91, 103 by air blast 88 forced oil circulation 106 water circulation 105 Copper losses 72, 75, 76, 83, 142, 165 resistivity of 144 space factor (see Winding space factor). Core loss (usual values) (sec also Losses in iron) 77 Core-type transformers 17, 22 Corrugations, effect of, on sides of tank 94 on insulator surface 60 Coulomb 34 Current density in windings 72 transformers 184 D Density (see Flux- and Current-density). Design coefficient (c) 149 numerical example in 154 problems 13 procedure in : 150 Dielectric circuit 32 constant 36 constants, table of 37 strengths, table of 37 Disruptive gradient 36, 62 Distributing transformers 17 E Eddy currents in copper windings 73 current losses (see Losses). Effective cooling surface of tanks 96 Efficiency. . . '. 73, 167 all-day 74 approximate, of commercial transformers 74, 183 calculation of, for any power factor 77 maximum 145 Elastance, definition 35 INDEX 207 PAGE Electrifying force 38 Electrostatic force 38 E.m.f. in transformer coils (see also Volts; Voltage) 4, 5, 6 Equivalent cooling surface of tanks 96 ohmic voltage drop. 134, 137 Exciting current 5, 125, 168 in auto-transformers 194 volt-amperes 129 (curves) 131 F Farad 33 Flux density, electrostatic 35 in transformer cores i. 72, 164 leakage (see Leakage flux). . Forces acting on transformer coils 24, 1 74 Frequency, effect of, on choice of iron . . 19 allowance for core loss 77 Furnaces, transformers for electric 177 H Heat conductivity of materials 80 copper 83, 87 insulation 87 Heating of transformers (see Temperature rise). High-voltage testing transformers 15 Hottest spot calculations 84 Hysteresis, losses due to (see Losses). I Induction regulator 197 polyphase 203 Instrument transformers 183 Insulation of end turns of transformer windings 50 oil 52 problems of transformer 32 thickness of 48 Iron, losses in 69, 77, 142, i8c 208 INDEX L PAGE Laminations, losses in 69, 77, 142, 189 shape of, in shell-type transformer 19 thickness of 19 Large transformers. . 16, 17 Leakage flux : 98, 107, 118, 179, 198 reactance (see Reactance; Reactive voltage drop). Losses, eddy current 69 hysteresis 69 in copper windings 72, 75, 76, 83, 142, 165 in iron circuit 69, 77, 142, 189 power, in transformers 69 M Magnetic leakage (see Leakage flux). Magnetization curves for transformer iron % 128 Magnetizing current (see Exciting current). Mechanical stresses in transformers 24, 1 74 Microfarad 36 O Oil insulation. , 52 Output equation 138 Overloads, effect of, on temperature 98 P Permeance 34, 39 Permittance (see Capacity). Polyphase transformers 12, 22 Potential gradient 38 Power losses (see Losses). transformers 16, 154 Q Quantity of electricity (Coulomb) 34 INDEX 209 R PAGE Reactance, leakage, experimental determination of 114 Reactive voltage drop 117, 137, 180 Regulation 109, 132, 168 formulas 134, 135 Regulating tranformers 197 polyphase 203 Reluctance, magnetic 35 Resistance of windings 165 thermal 81 Resistivity of copper 144 S "Sandwiched" coils 118 Saturation; reasons for avoiding high flux densities 164 Self-induction of secondary winding 108 Series, transformers 184 Shell-type transformers 17, 20, 24, 155 Short-circuited transformer, diagram of 116 Silicon-steel for transformer stampings 71 Single-phase units used for three-phase circuits 12 Space factor, copper (see Winding space factor) . iron 151 Sparking distance; in air 58, 68 in oil 52, 53 Specifications ' 140, 155 Specific inductive capacity (see Dielectric constant). heat; of copper 99 of oil 99 Stacking factor 151 Stampings, transformer, thickness of 19 Static shield on h.t. terminals 65, 68 Stresses in transformer coils 24, 1 74 Surface leakage 46 under oil 54 Symbols, list of.' ix 210 INDEX T PAGE Temperature rise of transformers 79, go, 92, 94, 98, 170 after overload of short duration 99 Terminals ; 54 composition-filled 59 condenser type 62 oil-filled 57, 60 porcelain 57 Test voltages. 58 Theory of transformer, elementary 2 Thermal conductivity (see Heat conductivity). ohm, definition 81 Three-phase transformers 12, 22 Transformers, auto 191 constant current 178 core-type 17, 20, 22 current 184 distributing 17 for electric furnaces 177 large currents 178 use with measuring instruments 183 polyphase 12, 22 power 16, 154 series : . . . . 184 shell-type 17, 20, 24, 155 welding 177 Tubular type of transformer tank 104 V Variable-ratio transformers. 197 Vector diagram illustrating effect of leakage flux no, 112 of auto-transformer 196' short-circuited transformer 1 16 series transformer 185, 186 transformer on inductive load n, i$$. 134, 135 non-inductive load . . .' 10 INDEX 211 PAGE Vector diagram of transformer with large amount of leakage flux. . . 180 open secondary circuit 5 variable leakage reactance 181 showing components of exciting current 126 Voltage, effect of, on design 15 drop due to leakage flux 117, 137 regulation (see Regulation). Voits per turn of winding < 141 W Water-cooled transformers 105 Weight of transformers 151, 173 Welding transformers 177 Windings, estimate of number of turns in 141 Winding space factor 51, 151, 152 :t Window," dimensions of, in shell-type transformers 160, 163 Wire, size of, in windings (see Current density). SUPPLEMENTARY INDEX OF TABLES, CURVES, AND FORMULAS A PAGE Air clearances (Formula) 49 quantity required for air-blast cooling 89, 90 Ampere-turns, allowance for joints 127 B B-H curves (Gausses and amp- turns per inch) 128 C Capacity current 42 in terms of dimensions, etc 36 Charging current 42 Cooling area of tanks (Curve) 93 Copper space factors 51, 151, 152 Core loss (usual values) 77 Corrugated tanks, correction factor for cooling surface of 96 Current density (usual values) 72 D Density, current, in coils (usual values) 72 in transformer cores (Table) 72 Dielectric constants (Table) 37 strengths (Table) 37 Disruptive gradient (see Dielectric strength). E Efficiency (usual values) 74 E.m.f., formulas 5, 6 214 SUPPLEMENTARY INDEX PAGE Equivalent surface of corrugated tanks (correction factor) 96 Exciting volt-amperes, Formula 130 Curve '. 131 Flux densities in core (Table) 72 Force exerted on coil by leakage flux 28 H Hottest spot tempe.-ature -(Formula) 86 Inductive voltage drop (Formula) 124 Insulation, air clearance 49 oil clearance . 53 thickness of (Table) 48 Iron loss (Curves) 70, 189 J Joints in iron circuit, ampere turns required for 127 Losses in cores (usual values) 77 transformer iron (Curves) 70, 189 M Magnetization curves for transformer iron 128 Magnetizing volt-amperes (Curve) 131 Mechanical force on coil due to magnetic field 28 O Oil, insulation thickness in 53, 54 transformer, test voltages 52 Output equation , 138 SUPPLEMENTARY INDEX 215 P PAGE Power losses in transformer iron (Curves) 70, 189 R Reactance, leakage, in terms of test data 117 Reactive voltage drop (Formula) 124 Regulation formulas 135, 136 Resistance, equivalent primary 117 S Space factors, copper 51, 151, 152 iron 151 Specific inductive capacity (Dielectric constant), (Table) ......... 37 Surface leakage distance, in air 50 under oil.. 54 T Temperature of hottest spot (Formula) 86 rise due to overloads (Formula) 98, 101 in terms of tank area (Curve) 93 Thickness of insulation 48 in oil 53, 54 V Voltage drop, reactive (Formula.) 124 regulation (Formulas) 135, 136 Volt-amperes of excitation, (P'ormula) 130 (Curves) 131 Volts per turn of winding (Formula) '. . 142 numerical constants 149 W Water, amount of, required for water-cooling coils 105 Winding space factors 51, 151, 152 216 SUPPLEMENTARY INDEX NUMERICAL EXAMPLES PAGE Capacities in series 43 Composition-filled bushing 58 Condenser-type bushing 65 Cooling-coil for water-cooled transformers 105, 171 "Hottest spot" temperature calculation 87 Layers of different insulation in series 44 Mechanical stresses in coils 1 74 Plate condenser 41 Temperature rise due to overloads 98, 102 of self cooling oil-immersed transformer 94 with tank having corrugated sides 97 Transformer design 154 Voltage regulation 136, 168 Volt-amperes of excitation per pound of iron in core 130 THIS BOOK IS DUE ON THE LAST DATE BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. ENGINEERIN LIBRARY :__ i o Aprs'* MAR 2 6 1948 AUG 27 1948 Ji(L_lflLI95J ^ftR AU6 22 1949 \ : 0y acri W V2 5 1949 10m-7,'44(10G4s)